Optimal reinsurance maximising dividends as an infinite-dimensional optimisation problem and numerical results

We formulate the optimal reinsurance problem maximising cumulative dividend payments in discrete-time, where our decision is the ceding loss function for each stage, given within a general family. For the surplus, reinsurance is applied to the aggregated loss of each stage and the reinsurance premium is given by a distortion risk measure. Considering dividends as part of the decision variable, we maximise our objective under (a) the surplus, and (b) adding a solvency constraint that controls the ruin probability. Thirdly, we solve a last problem, (c) under both constraints when dividends are given by a dividend rule. For (a)-(b), we find multi-layered optimal policies by minimising the expected loss of the insurer for each stage, moreover, (b) offers a dividend rule as a cap of the surplus. However, the policies of (a)-(b) are not practically justified unless the premium is calculated with coherent distortion risk measures, in which case it is optimal to not reinsure. The optimal policy for (c) with the barrier dividend rule can be found by solving a constrained problem for each stage, where the constraint imposes an upper bound to the retained losses. We obtain multi-layered policies, whose layers cannot be calculated as they depend on Lagrangian multipliers. We propose a Linear Programming (LP) to approximate these optimal policies. We show results for the Expected value, Value-at-Risk, Average-Value-at-Risk and Glue Value-at-Risk. The deductibles we estimate show a relationship with the distortion, the barrier, and the income of the insurer.