Optimal control of investment, premium and deductible for a non-life insurance company

Bent Jesper Christensen*, Juan Carlos Parra-Alvarez, Rafael Serrano

*Corresponding author for this work

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A risk-averse insurance company controls its reserve, modeled as a perturbed Cramér-Lundberg process, by choice of both the premium p and the deductible K offered to potential customers. The surplus is allocated to financial investment in a riskless and a basket of risky assets potentially correlating with the insurance risks and thus serving as a partial hedge against these. Assuming customers differ in riskiness, increasing p or K reduces the number of customers n(p,K) and increases the arrival rate of claims per customer λ(p,K) through adverse selection, with a combined negative effect on the aggregate arrival rate n(p,K)λ(p,K). We derive the optimal premium rate, deductible, investment strategy, and dividend payout rate (consumption by the owner-manager) maximizing expected discounted lifetime utility of intermediate consumption under the assumption of constant absolute risk aversion. Closed-form solutions are provided under specific assumptions on the distributions of size and frequency of claims.

Original languageEnglish
JournalInsurance: Mathematics and Economics
IssuePart B
Pages (from-to)384-405
Number of pages22
Publication statusPublished - Nov 2021


  • Adverse selection
  • Deductible control
  • Hamilton-Jacobi-Bellman equation
  • Jump-diffusion
  • Optimal investment strategy
  • Premium control
  • Stochastic optimal control


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