On exact solutions for dividend strategies of threshold and linear barrier type in a Sparre Andersen model.

*(English)*Zbl 1158.62071Summary: The problem of determining the optimal dividend strategy that maximizes the expected discounted dividend payments until ruin and at the same time does not lead to ruin with a specified probability bound is currently out of sight. An alternative approach is to formulate explicit strategies and to tune the involved parameters in such a way that a given target in terms of profitability and solvency is reached. For that approach explicit expressions in terms of the model parameters are necessary. We showed how to obtain such expressions for the Erlang renewal model. Using these explicit expressions, a comparison of a barrier strategy that evolves linearly in time with a threshold strategy (which depends on the surplus level) showed that the former can outperform the latter in terms of the usual optimization criteria (namely expected sum of discounted dividend payments versus survival probability). This shows that although a strategy that depends on time is perhaps less intuitive than one depending on the surplus level, it may be preferable in some situations, although most of the research literature is focussed on surplus-dependent strategies. In particular the results of this paper show that one has to be very careful about the choice of the dividend payout strategy.

Moreover, it should be kept in mind that even when ignoring the safety aspect and purely looking at expected profits, the higher moments of the discounted dividend payments under horizontal and threshold strategies in the literature can be considerably large, making the sole use of this optimization criterion somewhat questionable. One other way to improve upon that might be a utility-based approach. However, identifying corresponding optimal dividend strategies is in general extremely difficult.

Moreover, it should be kept in mind that even when ignoring the safety aspect and purely looking at expected profits, the higher moments of the discounted dividend payments under horizontal and threshold strategies in the literature can be considerably large, making the sole use of this optimization criterion somewhat questionable. One other way to improve upon that might be a utility-based approach. However, identifying corresponding optimal dividend strategies is in general extremely difficult.