Optimal Risk Transfers in Insurance Groups

Abstract

This is the accepted version of the paper.This version of the publication may differ from the final published version. Abstract. Permanent repository linkOptimal risk transfers are derived within an insurance group consisting of two separate legal entities, operating under potentially different regulatory capital requirements and capital costs. Consistent 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 minimising the risk-adjusted value of the total group liabilities, with valuation carried out using a cost-of-capital approach. The optimisation 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 that specifically arises 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.

Optimal Reinsurance Policies under the VaR Risk Measure When the Interests of Both the Cedent and the Reinsurer Are Taken into Account

“…Therefore, the Pareto-optimal reinsurance policies could be determined by minimizing a linear combination of the univariate VaRs of C f and R f . We note in this regard that the optimization criterion of minimizing linear combinations of the risks of the cedent and the reinsurer was adopted by [7,22]. Our arguments provide an additional economic meaning to such criteria.…”

“…Borch [6] showed that for a fixed premium and expected reinsurance payments, the variance of the cedent's losses is minimized by the excess-of-loss reinsurance policy. In recent years, various solutions to the optimal reinsurance problem have been obtained where the value-at-risk (VaR) and the tail-value-at-risk (TVaR) have been used to measure the cedent's risk level (e.g., [7][8][9][10][11][12][13] and the references therein).…”

“…The usual approach is to identify Pareto-optimal treaties, whereby no agent can be made better off without making another agent worse off. For results in this area, we refer to, e.g., [6,7,24,25] and the references therein.…”

Optimal Reinsurance with Heterogeneous Reference Probabilities

“…The insurer is enforced hold risk capital given by ρĝ I (X − f (X), P I ) − E P I [X − f (X)], where ρĝ I (X − f (X), P I ),ĝ I ∈ G, is a distortion risk measure used to determine the risk capital is based on the probability measure P I . Such a valuation principle is used commonly in practice and is embedded in regulatory requirements under the Swiss Solvency Test and Solvency II (see, e.g., Chi [25], Asimit et al [26], and Cheung and Lo [15]). The wealth at a pre-determined future time for the insurer is given bŷ…”

“…The reason for taking these two risk measures as the criteria for optimal reinsurance is due to their popularity among banks and insurance companies for quantifying risks and determining capital requirements. Asimit et al [26] study our setting with VaR and CVaR in the context of insurance risk transfers in more detail in case of a homogeneous reference probability.…”