Project CEMAPRE internal
| Title | Optimal reinsurance: constraints, smart reinsurance and emerging risks |
| Participants | João Andrade e Silva, Alfredo Egídio dos Reis, Manuel Guerra, Alexandra Moura (Principal Investigator), Carlos Oliveira, Onofre Simões |
| Summary | The project focuses on the optimal reinsurance problem, continuing the 2024 project with the title “optimal reinsurance with dependencies and applications”, together with new approaches and ideas for further studying risk modelling and its implications in insurance and reinsurance design. Besides the impact of constraints in the optimal reinsurance contract, and apart from considering dependences between the underlying risks, we will also analyse time dependences, such as in the Cox and Hawkes Processes, or in the “smart reinsurance” problem. We aim at concluding the study of the impact of constraints, such as the moral hazard constraint, in the optimal reinsurance treaty, and their relation with dependencies. From our previous studies, with the moral hazard constraint, the optimal reinsurance treaty of dependent risks may not be an increasing function of the underlying risk, which is often a requirement to avoid moral hazard. From our studies we already know that the moral hazard constraint, which is a Lipschitz type constraint, is an active constraint. Our purpose is now to characterise this constraints analytically and to obtain numerical illustrations through properly devised numerical algorithms. Further, we aim at continuing the analysis of the optimal surplus reinsurance treaty for a real insurer portfolio, started in 2023 in collaboration with a Portuguese insurance company. Indeed, regarding the surplus treaty in the optimal reinsurance problem, litle can be found in literature, which is mainly due to its complexity, as in spite being proportional reinsurance, the surplus treaty captures some properties of non-proportional reinsurance, making the optimisation problem significantly more complex. We study the problem considering background dependence using simulation. We will continue the study or We will continuing the collaboration with professor Hansjoerg Albrecher, from the University of Lausanne, in “smart reinsurance”. The goal is to study the problem of “smart” optimal reinsurance, in which the reinsurance contract can be changed during the term of the contract, leading to time dependences. This line of research, practically non-existent in published literature, is motivated by Professor Albrecher's contacts with important partners in the reinsurance industry. They expressed interest in the possibility of offering their customers contracts with greater flexibility. This increase in flexibility, together with time dependencies, makes the problem much more complex from a mathematical point of view. We also aim in this 2025 project to study the modelling of emerging risks, such as climate risk and cyber-risk, which are paramount for reinsurers, as they might lead to extreme losses often leading to activation of the reinsurance contract. One of the challenges of this emerging risks is that the number of claims is characterised by time dependence, that is, the intensity of the frequency of claims is not constant in time. Indeed, from the literature, it is understood that Cox-processes, in which the intensity of the Poisson process of the numer of claims is not constant and , are well suited for climate risk modelling, as the frequency of claims due to climate extreme events are well modelled by processes with shot-noise. It is also found in literature that the cyber-risk number of claims as the self-excitement characteristic, due for instance to human reaction, apart from shot-noise. Hawke’s process, which are further extensions of the Cox processes, account for this self-excitement increase in the intensity of the process, and they will be also considered to model risk in this study. |