Hanspeter Schmidli: Optimal Dividends and Capital Injection Problems in Non-Life Insurance

The traditional risk measure in actuarial mathematics is the ruin probability. This concept has been criticised because it does not take into account the time of ruin and the deficit at ruin. An alternative measure has been suggested by de Finetti (1957). He proposed to consider the discounted value of dividends paid from the portfolio. However, under the optimal dividend strategy ruin becomes certain. Moreover, the deficit at ruin is not taken into account, either. Moreover, ruin is certain under the optimal dividend strategy.

As an alternative, we allow capital injections that should keep the surplus positive. Ruin is not allowed in our model. In one model, we measure the risk as the value of the (discounted) capital injections. We look for the reinsurance strategy that minimises the value. A second model allows also dividend payments. Here, the value is the discounted dividends minus penalised capital injections. We show that the optimal dividend strategy is a barrier strategy.

This talk is based on joint work with Julia Eisenberg and Natalie Scheer.