Optimal Dividend and Capital Injection Problems in Non-Life Insurance
A talk in the Bielefeld Stochastic Afternoon series by
Hanspeter Schmidli from Köln
| Abstract: | 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 to 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.
As an alternative, we allow capital injections that have to 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. Discounting in these models has to be seen as a
preference measure: dividends today are preferred to dividends tomorrow and
injections tomorrow are preferred to injections today. Since the parameters
of the surplus process are kept constant, the new measures are also to be
considered as technical measures used for risk management.
This talk is based on joint work with Julia Eisenberg and Natalie Scheer. |