Optimal insurance design with a bonus

Yongwu Li, Zuoquan Xu

Research output: Journal article publicationJournal articleAcademic researchpeer-review

4 Citations (Scopus)


This paper investigates an insurance design problem, in which a bonus will be given to the insured if no claim has been made during the whole lifetime of the contract, for an expected utility insured. In this problem, the insured has to consider the so-called optimal action rather than the contracted compensation (or indemnity) due to the existence of the bonus. For any pre-agreed bonus, the optimal insurance contract is given explicitly and shown to be either the full coverage contract when the insured pays high enough premium, or a deductible one otherwise. The optimal contract and bonus are also derived explicitly if the insured is allowed to choose both of them. The contract turns out to be of either zero reward or zero deductible. In all cases, the optimal contracts are universal, that is, they do not depend on the specific form of the utility of the insured. A numerical example is also provided to illustrate the main theoretical results of the paper.
Original languageEnglish
Pages (from-to)111-118
Number of pages8
JournalInsurance: Mathematics and Economics
Publication statusPublished - 1 Nov 2017


  • Bonus–malus system
  • Expected utility
  • Insurance contract with bonus
  • Optimal insurance design
  • Personalized contract

ASJC Scopus subject areas

  • Statistics and Probability
  • Economics and Econometrics
  • Statistics, Probability and Uncertainty


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