Optimal investment-reinsurance with dynamic risk constraint and regime switching

Abstract

We study an optimal investment–reinsurance problem for an insurer who faces dynamic risk constraint in a Markovian regime-switching environment. The goal of the insurer is to maximize the expected utility of terminal wealth. Here the dynamic risk constraint is described by the maximal conditional Value at Risk over different economic states. The rationale is to provide a prudent investment–reinsurance strategy by taking into account the worst case scenario over different economic states. Using the dynamic programming approach, we obtain an analytical solution of the problem when the insurance business is modeled by either the classical Cramer–Lundberg model or its diffusion approximation. We document some important qualitative behaviors of the optimal investment–reinsurance strategies and investigate the impacts of switching regimes and risk constraint on the optimal strategies.

Keywords:

• optimal reinsurance and investment
• regime-switching
• utility maximization
• dynamic programming
• maximal conditional Value at Risk (MCVaR)
• regime-switching Hamilton–Jacobi–Bellman (HJB) equations

Acknowledgements

The authors would like to thank the reviewer for helpful comments. Tak Kuen Siu would like to acknowledge the Discovery Grant from the Australian Research Council (ARC), (Project No. DP1096243).

Notes

¹As pointed in Rockafellar & Uryasev (2000), to work directly with the function CVaR(Π _i , q), may be hard to do because of the nature of its definition in terms of the k-VaR value and the often troublesome mathematical properties of that value.