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 af dette arbejde

Institut for Økonomi

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

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.

Originalsprog	Engelsk
Tidsskrift	Insurance: Mathematics and Economics
Vol/bind	101
Nummer	Part B
Sider (fra-til)	384-405
Antal sider	22
ISSN	0167-6687
DOI	https://doi.org/10.1016/j.insmatheco.2021.07.005
Status	Udgivet - nov. 2021

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Citationsformater

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

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.",

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

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

note = "Publisher Copyright: {\textcopyright} 2021 Elsevier B.V.",

year = "2021",

month = nov,

doi = "10.1016/j.insmatheco.2021.07.005",

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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.
I: Insurance: Mathematics and Economics, Bind 101, Nr. Part B, 11.2021, s. 384-405.

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

TY - JOUR

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AU - Christensen, Bent Jesper

AU - Parra-Alvarez, Juan Carlos

AU - Serrano, Rafael

PY - 2021/11

Y1 - 2021/11

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

KW - Optimal investment strategy

KW - Premium control

KW - Stochastic optimal control

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