Analysis of Ruin Probability in Insurance Companies Using the Cramer-Lundberg Model with Variations in Claim Distributions

Authors

  • Alfi Khairiati Institut Sains dan Teknologi Nasional Jakarta
  • Retno Budiarti Department Mathematics, IPB University
  • Mohamad Khoirun Najib Department Mathematics, IPB University

DOI:

https://doi.org/10.30829/zero.v9i2.25581

Keywords:

Claim Distribution, Cramer-Lundberg, Monte Carlo, Ruin Probability

Abstract

Ruin risk is a critical concern in the insurance industry, reflecting a company's ability to fulfill long-term claim obligations. This study estimates ruin probabilities using the Cramer-Lundberg model, with Monte Carlo simulation applied to three claim distributions: exponential, lognormal, and gamma. Simulations vary initial capital while holding the premium rate, claim intensity, and distribution parameters constant. Results show that the lognormal distribution, due to its heavy tail, leads to higher ruin probabilities compared to exponential and gamma distributions. The gamma distributions produce intermediate outcomes, while the exponential shows the lowest risk. These findings highlight the importance of considering distributional characteristics when assessing solvency risk. The study provides practical insights for actuaries and risk managers in evaluating capital adequacy, stress-testing portfolios, and developing adaptive pricing and reinsurance strategies.

Author Biographies

Alfi Khairiati, Institut Sains dan Teknologi Nasional Jakarta

Department Matematika,  ISTN Jakarta, Indonesia 

Retno Budiarti, Department Mathematics, IPB University

Department Mathematics, IPB University

Mohamad Khoirun Najib, Department Mathematics, IPB University

Department Mathematics, IPB University

References

H. Albrecher, M. Bladt, and E. Vatamidou, "Efficient Simulation of Ruin Probabilities when claims are mixtures of heavy and light tails", Methodology and Computing in Applied Probability, vol.23, pp. 1237-1255, 2021. https://doi.org/10.1007/s11009-020-09799-6

J. Dhaene, R. J. A. Laeven, and Y. Zhang, "Systemic risk: Conditinal distortion risk measures", Insurance: Mathematics and Economics, vol.103, pp 126-145, 2022.

https://doi.org/10.1016/j.insmatheco.2021.12.002

A. Kuznetsov and Z. Palmowski, "Ruin probability in the classical Cramer-Lundberg model with investments," Insurance: Mathematics and Economics, vol. 108, pp 189-205, 2023. https://doi.org/10.1016/j.insmatheco.2022.12.004

Z. Cheng, and Y. Seol, "Diffusion Approximation of a Risk Model with Non-Stationary Hawkes Arrivals of Claims", Methodology Computing Appllied Probability, vol. 22, pp 555-571, 2020. https://doi.org/10.1007/s11009-019-09722-8

A. Bazyari, "Analysis of a Dependent Perturbed Renewal Risk Model with Heavy-tailed Distributions", Lobachevskii Journal of Mathematics, vol. 44, pp 4610-4629, 2023. https://doi.org/10.1134/S1995080223110057

A. Bazyari, "On the ruin probabilities for a general perturbed renewal risk process," Journal of Statistical Planning and Inference, vol. 227, pp 1-17, 2023. https://doi.org/10.1016/j.jspi.2023.02.005

O. Melnikov, and J. Milz, "Randomized Quasi-Monte Carlo Methods for Risk-Averse Stochastic Optimization", Journal of Optimization Theory and Applications, vol. 206, pp 14, 2025. https://doi.org/10.1007/s10957-025-02693-6

M. Tomita, K. Takaoka, and M, Ishizaka, "Some mathematical properties of the premium function and ruin probability of a generalized Cramer-Lundberg model driven by mixed poisson processes", Japan Journal of Industrial and Applied Mathematics, vol. 41, pp 1389-1412, 2024. https://doi.org/10.1007/s13160-024-00656-4

Swiss Re Institute, "Advanced Risk Modelling for Reinsurance Strategies", Swiss Re, 2022. [Online]. Available: https://www.swissre.com/institute

A. Alfonsi, A. Cherchali, J. A. I. Acevedom, " Multilevel Monte Carlo for computing the SCR with the standard formula and other stress tests," Insurance: Mathematics and Economics, vol.100, pp 235-260, 2021. https://doi.org/10.1016/j.insmatheco.2021.05.005

F. Perla, S. Scognamiglio, A. Spadaro, and P. Zanetti, "Explainable Least Square Monte Carlo for Solvency Capital Requirement Evaluation," North Americal Actuarial Journal, 2025. https://doi.org/10.1080/10920277.2025.2519542

S. Graf, and R. Korn, "A guide to Monte Carlo simulation concepts for assessment of risk-return profiles for regulatory purposes," European Actuarial Journal, vol.10, pp 273-293, 2020. https://doi.org/10.1007/s13385-020-00232-3

S. A. Klugman, H. H. Panjer, and G. E. Willmot, Loss Models: From Data to Decisions, 4th ed. Hoboken, NJ: Wiley, 2012. http://dx.doi.org/10.1002/9780470391341.index

G. Willmot and J. K. Woo, Surplus Analysis of Sparre Andersen Insurance Risk Processes. Springer, New York, 2017. https://doi.org/10.1007/978-3-319-71362-5

T. Dimitrakopoulou, A. Karagrigoriou, A. Makrides, I. Vonta, "Competing Risks Modelling via Multistate Sysem Methodology under a Generalized Family of Distributions," Methodology and Computing in Applied Probability, vol. 27:41, 2025. https://doi.org/10.1007/s11009-025-10169-3

D. Gaigall, "Test for Changes in The Modeled Solvency Capital Requirement of An Internal Risk Model", Journal of the IAA, vol. 51, pp 813-837, 2021. https://doi.org/10.1017/asb.2021.20

Y. K. Tse, "Basic Monte Carlo Methods," Nonlife Actuarial Models, Singapore Management University, pp 370-401, 2023. https://doi.org/10.1017/9781009315067.014

J. Tom, The Cramer-Lundberg Model and Copulas, Eindhoven University of Technology, Thesis, 2021.

M. C. Fu, Simulation and the Monte Carlo Method, 3rd ed., Hoboken, NJ: Wiley, 2023. https://doi.org/10.1002/9781118631980

M. Mandjes and O. Boxma, The Cramer-Lundberg model and its variants, Spinger Nature, 2023. https://doi.org/10.1007/978-3-031-39105-7

T. M. Tovstik, and D. S. Bulgakova, "An Insurance Company Model with Random Premium and Claims", Vestnik St.Petersb Univ.Math, vol. 58, pp 79-91, 2025. https://doi.org/10.1134/S1063454125700098

EIOPA, "Solvency II: 2023 Insurance Stress Test Report," European Insurance and Occupational Pensions Authority, 2023. [Online]. Available: https://www.eiopa.europa.eu

Society of Actuaries, "Monte Carlo Simulation Application in Insurance Risk Management", SOA Research Report, 2024. [Online]. Available: https://www.soa.org/resources/research-reports/2024/monte-carlo-simulation-insurance-risk

D. C. M. Dickson, M. R. Hardy, dan H. R. Waters, Actuarial Mathematics for Life Contingent Risks, 2nd ed., Cambridge: Cambrige University Press, 2013. https://doi.org/10.1017/9781108784184

J. Grandell, Aspects of Risk Theory, Springer Science & Business Media, 2012. https://doi.org/10.1007/978-1-4613-9058-9

Downloads

Additional Files

Published

2025-10-10

Issue

Vol. 9 No. 2 (2025): Zero: Jurnal Sains Matematika dan Terapan

Section

Articles

License

Authors who publish with this journal agree to the following terms:

  1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution-ShareAlike 4.0 International License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).