A quasi-conjugate bivariate prior distribution suitable for studying dependence in reinsurance and non reinsurance models with and without a layer

Emilio Gómez-Déniz; Emilio Gómez-Déniz

doi:10.3934/math.2025546

AIMS Mathematics

2025, Volume 10, Issue 5: 12055-12078. doi: 10.3934/math.2025546

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A quasi-conjugate bivariate prior distribution suitable for studying dependence in reinsurance and non reinsurance models with and without a layer

Emilio Gómez-Déniz ^,

Department of Quantitative Methods and TIDES Institute, University of Las Palmas de Gran Canaria, Campus de Tafira s/n, 35017-Las Palmas de Gran Canaria, Spain

Received: 15 January 2025 Revised: 05 May 2025 Accepted: 16 May 2025 Published: 26 May 2025
MSC : 62C10, 91B05, 91G05

In the collective risk model and also in a compound excess-of-loss reinsurance frameworks, it is usual to assume that the risk parameters associated with the random variables, the number of claims, and the claim size are independent for mathematical convenience. Here, we assumed Poisson and Pareto distributions for these two random variables. This paper focuses on the prior and posterior (Bayesian) net premiums of the total claims amount, assuming some degree of dependence between the two risk profiles associated with these two random variables. Here, the degree of dependence was modeled using the Sarmanov-Lee family of distributions, a special type of copula, which allows us to study the impact of this assumption on the following year's total cost of claims when prior margins are assumed to have gamma and shifted Erlang distributions. The numerical applications show that a low degree of correlation between these variables leads to collective and net Bayes premiums that can be sensitive when the hypothesis of independence is broken. The dependence hypothesis has a more significant effect in the model when no layer is considered. We illustrate the methodology proposed with some real numerical examples.
- Bayes,
- dependence,
- excess-of-loss,
- layer,
- risk parameters,
- Sarmanov-Lee family of distributions
Citation: Emilio Gómez-Déniz. A quasi-conjugate bivariate prior distribution suitable for studying dependence in reinsurance and non reinsurance models with and without a layer[J]. AIMS Mathematics, 2025, 10(5): 12055-12078. doi: 10.3934/math.2025546

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Abstract

In the collective risk model and also in a compound excess-of-loss reinsurance frameworks, it is usual to assume that the risk parameters associated with the random variables, the number of claims, and the claim size are independent for mathematical convenience. Here, we assumed Poisson and Pareto distributions for these two random variables. This paper focuses on the prior and posterior (Bayesian) net premiums of the total claims amount, assuming some degree of dependence between the two risk profiles associated with these two random variables. Here, the degree of dependence was modeled using the Sarmanov-Lee family of distributions, a special type of copula, which allows us to study the impact of this assumption on the following year's total cost of claims when prior margins are assumed to have gamma and shifted Erlang distributions. The numerical applications show that a low degree of correlation between these variables leads to collective and net Bayes premiums that can be sensitive when the hypothesis of independence is broken. The dependence hypothesis has a more significant effect in the model when no layer is considered. We illustrate the methodology proposed with some real numerical examples.

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