A Novel Approximation Method for Computing the Adjustment Coefficient of a Nonlinear Cramér-Lundberg Risk Model with Gamma Claims

This study considers a non-linear Cramér-Lundberg risk model and examines the adjustment coefficient ( r ) when the claims have gamma distribution. The linear models are not always adequate because an insurance company’s premium income does not always increase linearly. Therefore, in this study, a m...

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Published in	Methodology and computing in applied probability Vol. 27; no. 3; p. 64
Main Authors	Gever Ekinci, Basak, Hanalioglu, Zulfiye, Khaniyev, Tahir
Format	Journal Article
Language	English
Published	New York Springer US 01.09.2025 Springer Nature B.V
Subjects	Applied mathematics Business and Management Economics Electrical Engineering Integral equations Life Sciences Mathematics and Statistics Probability Probability distribution functions Random variables Risk Statistical analysis Statistics Stochastic models Upper bounds Ruin probability Gamma distribution Non-linear Cramér-Lundberg risk model 62P05 91G60 Approximate formula for adjustment coefficient
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Abstract	This study considers a non-linear Cramér-Lundberg risk model and examines the adjustment coefficient ( r ) when the claims have gamma distribution. The linear models are not always adequate because an insurance company’s premium income does not always increase linearly. Therefore, in this study, a more realistic non-linear Cramér-Lundberg risk model is mathematically constructed. Then, the ruin probability of this non-linear risk model is studied when the premium function is in the form of square root function, i.e., p ( t ) = c t . It leads to analyzing the adjustment coefficient ( r ) , as examining this coefficient is required for finding an upper bound while investigating the ruin probability. However, in general case, it is a challenging procedure to calculate the exact value of r from an integral equation. Thus, in this study, the adjustment coefficient r is explored by computational methods and a new approximate formula for the practical calculation of the adjustment coefficient is proposed. Moreover, an implementation of the obtained approximate formula, which investigates ruin probability, is included as an example at the end of the paper.
AbstractList	This study considers a non-linear Cramér-Lundberg risk model and examines the adjustment coefficient ( r ) when the claims have gamma distribution. The linear models are not always adequate because an insurance company’s premium income does not always increase linearly. Therefore, in this study, a more realistic non-linear Cramér-Lundberg risk model is mathematically constructed. Then, the ruin probability of this non-linear risk model is studied when the premium function is in the form of square root function, i.e., p ( t ) = c t . It leads to analyzing the adjustment coefficient ( r ) , as examining this coefficient is required for finding an upper bound while investigating the ruin probability. However, in general case, it is a challenging procedure to calculate the exact value of r from an integral equation. Thus, in this study, the adjustment coefficient r is explored by computational methods and a new approximate formula for the practical calculation of the adjustment coefficient is proposed. Moreover, an implementation of the obtained approximate formula, which investigates ruin probability, is included as an example at the end of the paper. This study considers a non-linear Cramér-Lundberg risk model and examines the adjustment coefficient (r) when the claims have gamma distribution. The linear models are not always adequate because an insurance company’s premium income does not always increase linearly. Therefore, in this study, a more realistic non-linear Cramér-Lundberg risk model is mathematically constructed. Then, the ruin probability of this non-linear risk model is studied when the premium function is in the form of square root function, i.e., p(t)=ct. It leads to analyzing the adjustment coefficient (r), as examining this coefficient is required for finding an upper bound while investigating the ruin probability. However, in general case, it is a challenging procedure to calculate the exact value of r from an integral equation. Thus, in this study, the adjustment coefficient r is explored by computational methods and a new approximate formula for the practical calculation of the adjustment coefficient is proposed. Moreover, an implementation of the obtained approximate formula, which investigates ruin probability, is included as an example at the end of the paper.
ArticleNumber	64
Author	Khaniyev, Tahir Gever Ekinci, Basak Hanalioglu, Zulfiye
Author_xml	– sequence: 1 givenname: Basak surname: Gever Ekinci fullname: Gever Ekinci, Basak email: bgever@thk.edu.tr organization: Department of Industrial Engineering, University of Turkish Aeronautical Association – sequence: 2 givenname: Zulfiye surname: Hanalioglu fullname: Hanalioglu, Zulfiye organization: Department of Actuarial Sciences, Karabuk University – sequence: 3 givenname: Tahir surname: Khaniyev fullname: Khaniyev, Tahir organization: Department of Industrial Engineering, TOBB University of Economics and Technology, The Center of Digital Economics, Azerbaijan State University of Economics
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SubjectTerms	Applied mathematics Business and Management Economics Electrical Engineering Integral equations Life Sciences Mathematics and Statistics Probability Probability distribution functions Random variables Risk Statistical analysis Statistics Stochastic models Upper bounds
Title	A Novel Approximation Method for Computing the Adjustment Coefficient of a Nonlinear Cramér-Lundberg Risk Model with Gamma Claims
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