Optimal Risk Transfers in Insurance Groups

Abstract

Optimal risk transfers are derived within an insurance group consisting of two separate legal entities, operating under potentially different regulatory capital requirements and capital costs. Consistently with regulatory practice, capital requirements for each entity are computed by either a Value-at-Risk or an Expected Shortfall risk measure. The optimality criterion consists of minimizing the risk-adjusted value of the total group liabilities, with valuation carried out using a cost-of-capital approach. The optimization problems are analytically solved and it is seen that optimal risk transfers often involve the transfer of tail risk (unlimited reinsurance layers) to the more weakly regulated entity. We show that, in the absence of a capital requirement for the credit risk arising from the risk transfer, optimal risk transfers achieve capital efficiency at the cost of increasing policyholder deficit. However, when credit risk is properly reflected in the capital requirement, incentives for tail-risk transfers vanish and policyholder welfare is restored.

Keywords: Cost of Capital, Expected Shortfall, Insurance Groups, Optimal Reinsurance, Value-at-Risk