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

Bent Jesper Christensen; Juan Carlos Parra-Alvarez; Rafael Serrano

doi:10.1016/j.insmatheco.2021.07.005

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

Department of Economics and Business Economics

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

6 Citations (Scopus)

41 Downloads (Pure)

Abstract

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 language	English
Journal	Insurance: Mathematics and Economics
Volume	101
Issue	Part B
Pages (from-to)	384-405
Number of pages	22
ISSN	0167-6687
DOIs	https://doi.org/10.1016/j.insmatheco.2021.07.005
Publication status	Published - Nov 2021

Keywords

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

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abstract = "A risk-averse insurance company controls its reserve, modeled as a perturbed Cram{\'e}r-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.",

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author = "Christensen, {Bent Jesper} and Parra-Alvarez, {Juan Carlos} and Rafael Serrano",

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Optimal control of investment, premium and deductible for a non-life insurance company. / Christensen, Bent Jesper ; Parra-Alvarez, Juan Carlos; Serrano, Rafael.
In: Insurance: Mathematics and Economics, Vol. 101, No. Part B, 11.2021, p. 384-405.

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

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AU - Christensen, Bent Jesper

AU - Parra-Alvarez, Juan Carlos

AU - Serrano, Rafael

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N2 - 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.

AB - 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.

KW - Adverse selection

KW - Deductible control

KW - Hamilton-Jacobi-Bellman equation

KW - Jump-diffusion

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KW - Stochastic optimal control

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ER -

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

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