The Design of Optimal Re-Insurance Contracts when Losses are Clustered

This paper investigates the form of optimal reinsurance contracts in the case of clusters of losses. The underlying insured risk is represented by a marked Hawkes process, where the intensity of the jumps depends not only on the occurrence of previous jumps but also on the size of the jumps, which represents the financial magnitude of the loss. The reinsurance contracts are applied to each loss at the time of occurrence, but their structure is assumed to be constant. We derive closed-form formulas within the meanvariance framework. Additionally, we demonstrate that the optimal contract is not the classical excess-loss (deductible) form. The optimal contract is piecewise linear with three ranges: first, no reinsurance below a certain threshold; second, reinsurance with a slope greater than 1; and finally, full reinsurance. When the marked process converges to a Poisson process, we recover the optimality of the deductible form.