Estimating Reinsurance Premiums Using Pareto Conjugate Priors and Extreme Value Methods: Studies Case of Fire Insurance Claims in Denmark

Putri Isnaini Cahyaning Baiti; Adhitya Ronnie Effendie

doi:10.20956/j.v22i1.45510

Authors

Putri Isnaini Cahyaning Baiti Actuarial Program Study, Faculty of Science, Institute Technology Sumatera, Indonesia
Adhitya Ronnie Effendie Actuarial Program Study, Faculty of Science, Institute Technology Sumatera, Indonesia

DOI:

https://doi.org/10.20956/j.v22i1.45510

Keywords:

Loss Distribution, Bayesian Estimation, Premium Reinsurance, Extreme Value Theory

Abstract

Loss distributions in insurance are typically right-skewed with heavy tails. As a result, modelling such distributions often involves the use of heavy-tailed distributions, such as the Pareto family, Cauchy, Student-t, and mixture distributions. This study employs the Generalized Inverse Gaussian (GIG) distribution as a conjugate prior to the Pareto distribution. The GIG distribution is characterized by three parameters and includes the modified Bessel function of the third kind in its density, which makes parameter estimation using the likelihood method challenging. Therefore, a Bayesian estimation approach is adopted, utilizing two prior distributions from the GIG family: the Inverse Gaussian and the Reciprocal Inverse Gaussian. The modelling is carried out within the framework of Extreme Value Theory (EVT), focusing on excess values over a specified threshold and the probability of claims exceeding that threshold. The results obtained from this analysis can be used to derive a premium estimation formula that insurance companies can apply when reinsuring their claims with a reinsurance company

References

[1] Albrecher, H. Jan B. & Jozef L.T., 2017. Reinsurance: Actuarial and Statistics Aspects. John Wiley & Son ltd., United Kingdom.

[2] Charpentier, A., 2014. Computational Actuarial Science with R. Chapman & Hall/CRC., Canada.

[3] Cooray, K. & Malwane, M.A.A., 2009. Modelling Actuarial Data with Composite Lognormal-Pareto Model. Scandinavian Actuarial Journal, Vol. 5, 321-334.

[4] Deniz, E.G., 2025. A Quasi-Conjugate Bivariate Prior Distribution Suitable for Studying Dependence in Reinsurance and Non Reinsurance Models with and without a Layer. AIMS Mathematics, Vol 10, No. 5, 12055-12078.

[5] Embrechts, P., 1997. Modelling Extremal Event for Insurance and Finance. Springer., London.

[6] Gaigall, D. & Julian G., 2025. Fixed values versus empirical quantiles as thresholds in excess distribution modelling. Journal of Statistical Planning and Inference, Vol. 238, 106276.

[7] Geogebeur, Y. Armelle G. & Jing Q., 2021. Extreme Value Estimation of the Conditional Risk Premium in Reinsurance. Insurance: Mathematics and Economomics Journal, Vol. 96, 68-80.

[8] Gosh, S. & Sidney R., 2010. A Discussion on Mean Excess Plot. Stocastics Processes and Their Applications, Vol. 120, No. 8, 1492-1517.

[9] Haan, L.D. & Ana, F., 2006. Extreme Value Theory An Introduction. Springer Science+Business Media., United States.

[10] Hesselager, O., 1993. A Class of Conjugate Priors with Applications to Excess of Loss Reinsurance. Astin Bulletin, Vol. 23, No. 1, 77-93.

[11] Jorgensen, B., 1982. Statistical Properties of the Generalized Inverse Gaussian Distribution. Springer-Verlag., New York.

[12] Klugman, S.A. Harry H.P. & Gordon E.W., 2012. Loss Models from Data to Decisions Fourth Edition. John Wiley & Son., New Jersey.

[13] Liang, X. Zhibin L. & Virginia R.Y., 2020. Optimal Reinsurance under the Mean-Variance Premium Principle to Minimize the Probability of Ruin. Insurance: Mathematics and Economics Journal, Vol. 92, 128146.

[14] Mehr, R.I. Cammack E. & Rose T., 1985. Principle of Insurance. Eight Edition. University of Michigan, Michigan.

[15] Peng, X. & Yankai W., 2025. Alpha-Robust Mean-Variance Reinsurance and Investment Strategies with Transaction Costs. Journal Computational and Applied Mathematics, Vol. 457, 116257.

[16] Reiss, R.D. & Thomas M., 1999. A New Class of Bayesian Estimators in Paretian Excess-of-Loss Reinsurance. Astin Bulletin, Vol. 29, No. 2, 339-349.

[17] Subanar. 2013, Statistika Matematika. Graha Ilmu, Yogyakarta.

[18] Zanon, L.J.V. & Cristina, L.C., 2007. On Pareto Conjugate Priors and Their Application to Large Claims Reinsurance Premium Calculation. Astin Bulletin, Vol. 37, No. 2, 405-428.