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

Research output: Contribution to journal › Journal article › Research › peer-review

5 Citations (Scopus)

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
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
Externally published	Yes

Bibliographical note

Funding Information:
Bent Jesper Christensen and Juan Carlos Parra-Alvarez are also research fellows at the Danish Finance Institute (DFI), the Center for Research in Econometric Analysis of TimE Series (CREATES), and the Dale T. Mortensen Center (DTMC). Part of this research was done while Rafael Serrano was visiting Aarhus University, and he gratefully acknowledges financial support from DTMC . We are also grateful to the anonymous reviewers for helpful comments.

Funding Information:
Bent Jesper Christensen and Juan Carlos Parra-Alvarez are also research fellows at the Danish Finance Institute (DFI), the Center for Research in Econometric Analysis of TimE Series (CREATES), and the Dale T. Mortensen Center (DTMC). Part of this research was done while Rafael Serrano was visiting Aarhus University, and he gratefully acknowledges financial support from DTMC. We are also grateful to the anonymous reviewers for helpful comments.

Publisher Copyright:
© 2021 Elsevier B.V.

Keywords

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

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@article{3827d826fdff4e57b83ef379c6ee4859,

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 = "Funding Information: Bent Jesper Christensen and Juan Carlos Parra-Alvarez are also research fellows at the Danish Finance Institute (DFI), the Center for Research in Econometric Analysis of TimE Series (CREATES), and the Dale T. Mortensen Center (DTMC). Part of this research was done while Rafael Serrano was visiting Aarhus University, and he gratefully acknowledges financial support from DTMC . We are also grateful to the anonymous reviewers for helpful comments. Funding Information: Bent Jesper Christensen and Juan Carlos Parra-Alvarez are also research fellows at the Danish Finance Institute (DFI), the Center for Research in Econometric Analysis of TimE Series (CREATES), and the Dale T. Mortensen Center (DTMC). Part of this research was done while Rafael Serrano was visiting Aarhus University, and he gratefully acknowledges financial support from DTMC. We are also grateful to the anonymous reviewers for helpful comments. Publisher Copyright: {\textcopyright} 2021 Elsevier B.V.",

year = "2021",

month = nov,

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

language = "English",

volume = "101",

pages = "384--405",

journal = "Insurance: Mathematics and Economics",

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AU - Christensen, Bent Jesper

AU - Parra-Alvarez, Juan Carlos

AU - Serrano, Rafael

N1 - Funding Information: Bent Jesper Christensen and Juan Carlos Parra-Alvarez are also research fellows at the Danish Finance Institute (DFI), the Center for Research in Econometric Analysis of TimE Series (CREATES), and the Dale T. Mortensen Center (DTMC). Part of this research was done while Rafael Serrano was visiting Aarhus University, and he gratefully acknowledges financial support from DTMC . We are also grateful to the anonymous reviewers for helpful comments. Funding Information: Bent Jesper Christensen and Juan Carlos Parra-Alvarez are also research fellows at the Danish Finance Institute (DFI), the Center for Research in Econometric Analysis of TimE Series (CREATES), and the Dale T. Mortensen Center (DTMC). Part of this research was done while Rafael Serrano was visiting Aarhus University, and he gratefully acknowledges financial support from DTMC. We are also grateful to the anonymous reviewers for helpful comments. Publisher Copyright: © 2021 Elsevier B.V.

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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DO - 10.1016/j.insmatheco.2021.07.005

M3 - Journal article

AN - SCOPUS:85113785093

SN - 0167-6687

VL - 101

SP - 384

EP - 405

JO - Insurance: Mathematics and Economics

JF - Insurance: Mathematics and Economics

ER -

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

Abstract

Bibliographical note

Keywords

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