Optimal reinsurance under a new design: two layers and multiple reinsurers

We investigate the optimal reinsurance problem considering a more realistic two-layer design involving excess-of-loss reinsurance and multiple reinsurers. Unlike the commonly assumed single-layer design in the existing literature, our approach involves a top layer consisting of excess-of-loss reinsurance and the involvement of multiple reinsurers. Our focus is on the insurer's management of risk exposure and costly capital reinvestment strategies to maximize the company value. We formulate this problem as an optimization problem with constrained multivariate reinsurance exposure variables, and capital injection control. Our findings show that, from a profitability standpoint, when the surplus is low and the excess-of-loss reinsurance is not too expensive, the insurer should obtain a two-layer reinsurance with an excess-of-loss reinsurance at the top layer and a combination of proportional reinsurance from multiple reinsurers at the bottom layer. In contrast, when the surplus is relatively large or the excess-of-loss reinsurance is too expensive, the insurer should purchase a single-layer proportional reinsurance from multiple reinsurers. When the surplus is large enough, the insurer should not acquire any reinsurance. Our results suggest that stand-alone excess-of-loss reinsurance is never optimal when there is also proportional reinsurance with a variance premium principle available for losses below the excess threshold, and we theoretically confirm that a multi-layer reinsurance design is preferable over a single-layer design. Furthermore, our results demonstrate that in order to maximize the company's value, dividends should be paid according to the lump-sum barrier strategy, and capital injections should be considered only if the surplus is null and the transaction costs on capital injections are not too high.