Enhancing an insurer's expected value by reinsurance and external financing

In this paper, we analyze a decision-making problem for an insurer with limited liability, who is subject to a solvency constraint and wants to maximize the expected value through reinsurance purchase and external financing. We impose mild conditions on the reinsurance premium principle, which are the axioms of law invariance, risk loading and preserving the convex order. These three axioms are satisfied by all the widely used premium principles, except the Esscher principle, listed in Young (2004). If value at risk (VaR) or conditional value at risk (CVaR) is adopted to calculate the insurer's regulatory capital, we show that the optimal reinsurance can be in the form of two layers. The optimal reinsurance form is reduced to one layer once the premium principle further satisfies a weak condition. In particular, under Wang's, expected value, variance and Dutch premium principles, we derive the insurer's optimal strategies of reinsurance and financing, which are obtained explicitly under the VaR risk measure but have to be solved numerically for the CVaR risk measure. Results indicate that the insurer has three types of optimal strategies: the reinsurance only strategy, external financing only strategy, and mixed strategy. Moreover, the insurer's attitude toward ceding the risk through reinsurance and increasing capital through external financing is often greatly affected by the regulatory regime, the financing cost and the reinsurance price.