Optimal investment and reinsurance policies for the Cramér–Lundberg risk model under monotone mean-variance preference

Abstract

In this paper, an optimisation problem for the monotone mean-variance (MMV) criterion is considered from the perspective of the insurance company. The MMV criterion is an amended version of the classical mean-variance (MV) criterion which guarantees the monotonicity of the utility function. With this criterion we study the optimal investment and reinsurance problem which is formulated as a zero-sum game between the insurance company and an imaginary player. We apply the dynamic programming principle to obtain the corresponding Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation. As the main conclusion of this paper, by solving the HJBI equation explicitly, the closed forms of the optimal strategy and the value function are obtained. Moreover, the MMV efficient frontier is also provided. At the end of the paper, a numerical example is presented.

Keywords:

• Optimal reinsurance
• monotone mean-variance preference
• zero-sum game
• Hamilton–Jacobi–Bellman–Isaacs equation
• monotone efficient frontier

JEL classifications::

• C61
• G11
• G22

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant numbers 11931018, 12271274, and 12201104], the Tianjin Natural Science Foundation [grant number 19JCYBJC30400] and the Fundamental Research Funds for the Central Universities [grant number 2232021D-29].