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Two stackelberg games in life insurance: Mean-variance criterion

Part of: Game theory Hamilton-Jacobi theories, including dynamic programming Mathematical finance

Published online by Cambridge University Press:  23 December 2024

Xiaoqing Liang Open the ORCID record for Xiaoqing Liang [Opens in a new window]  and
Virginia R. Young Open the ORCID record for Virginia R. Young [Opens in a new window]
Xiaoqing Liang*
Affiliation:
School of Sciences, Hebei University of Technology, Tianjin, 300401, P. R. China
Virginia R. Young
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA
*
Corresponding author: Xiaoqing Liang; Email: [email protected]
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Abstract

We study two continuous-time Stackelberg games between a life insurance buyer and seller over a random time horizon. The buyer invests in a risky asset and purchases life insurance, and she maximizes a mean-variance criterion applied to her wealth at death. The seller chooses the insurance premium rate to maximize its expected wealth at the buyer’s random time of death. We consider two life insurance games: one with term life insurance and the other with whole life insurance—the latter with pre-commitment of the constant investment strategy. In the term life insurance game, the buyer chooses her life insurance death benefit and investment strategy continuously from a time-consistent perspective. We find the buyer’s equilibrium control strategy explicitly, along with her value function, for the term life insurance game by solving the extended Hamilton–Jacobi–Bellman equations. By contrast, in the whole life insurance game, the buyer pre-commits to a constant life insurance death benefit and a constant amount to invest in the risky asset. To solve the whole life insurance problem, we first obtain the buyer’s objective function and then we maximize that objective function over constant controls. Under both models, the seller maximizes its expected wealth at the buyer’s time of death, and we use the resulting optimal life insurance premia to find the Stackelberg equilibria of the two life insurance games. We also analyze the effects of the parameters on the Stackelberg equilibria, and we present some numerical examples to illustrate our results.

Keywords

Stackelberg gameterm life insurancewhole life insurancetime-consistent equilibriummean-variance criterion

MSC classification

Primary: 91A05: 2-person games
Secondary: 91A15: Stochastic games 91G10: Portfolio theory 49L20: Dynamic programming method
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 55 , Issue 1 , January 2025 , pp. 178 - 203
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The International Actuarial Association

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