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On the Moments of the Aggregate Discounted Claims with Dependence Introduced by a FGM Copula

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  • Mathieu Bargès

    (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon, Ecole d'Actuariat - ULaval - Université Laval [Québec])

  • Hélène Cossette

    (Ecole d'Actuariat - ULaval - Université Laval [Québec])

  • Stéphane Loisel

    (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

  • Etienne Marceau

    (Ecole d'Actuariat - ULaval - Université Laval [Québec])

Abstract

In this paper, we investigate the computation of the moments of the compound Poisson sums with discounted claims when introducing dependence between the interclaim time and the subsequent claim size. The dependence structure between the two random variables is defined by a Farlie-Gumbel-Morgenstern copula. Assuming that the claim distribution has finite moments, we give expressions for the first and the second moments and then we obtain a general formula for any mth order moment. The results are illustrated with applications to premium calculation, moment matching methods, as well as inflation stress scenarios in Solvency II.

Suggested Citation

  • Mathieu Bargès & Hélène Cossette & Stéphane Loisel & Etienne Marceau, 2011. "On the Moments of the Aggregate Discounted Claims with Dependence Introduced by a FGM Copula," Post-Print hal-00426502, HAL.
    • Handle: RePEc:hal:journl:hal-00426502
    Note: View the original document on HAL open archive server: https://hal.science/hal-00426502
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    Cited by:

    1. Ghislain Léveillé & Ilie-Radu Mitric & Victor Côté, 2018. "Effects of the Age Process on Aggregate Discounted Claims," Risks, MDPI, vol. 6(4), pages 1-17, September.
    2. Fouad Marri & Franck Adékambi & Khouzeima Moutanabbir, 2018. "Moments of Compound Renewal Sums with Dependent Risks Using Mixing Exponential Models," Risks, MDPI, vol. 6(3), pages 1-17, August.
    3. Siti Norafidah Mohd Ramli & Jiwook Jang, 2014. "Neumann Series on the Recursive Moments of Copula-Dependent Aggregate Discounted Claims," Risks, MDPI, vol. 2(2), pages 1-16, May.
    4. Woo, Jae-Kyung & Cheung, Eric C.K., 2013. "A note on discounted compound renewal sums under dependency," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 170-179.
    5. repec:hal:wpaper:hal-00735843 is not listed on IDEAS
    6. Albrecher, Hansjörg & Cheung, Eric C.K. & Liu, Haibo & Woo, Jae-Kyung, 2022. "A bivariate Laguerre expansions approach for joint ruin probabilities in a two-dimensional insurance risk process," Insurance: Mathematics and Economics, Elsevier, vol. 103(C), pages 96-118.
    7. Lefèvre, Claude & Picard, Philippe, 2011. "A new look at the homogeneous risk model," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 512-519.
    8. Shuanming Li & Yi Lu, 2018. "On the Moments and the Distribution of Aggregate Discounted Claims in a Markovian Environment," Risks, MDPI, vol. 6(2), pages 1-16, May.
    9. Li, Yaohan & Dong, You & Guo, Hongyuan, 2023. "Copula-based multivariate renewal model for life-cycle analysis of civil infrastructure considering multiple dependent deterioration processes," Reliability Engineering and System Safety, Elsevier, vol. 231(C).
    10. Blier-Wong, Christopher & Cossette, Hélène & Marceau, Etienne, 2023. "Risk aggregation with FGM copulas," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 102-120.
    11. Zhao, Xiaobing & Zhou, Xian, 2012. "Copula models for insurance claim numbers with excess zeros and time-dependence," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 191-199.
    12. Marri, Fouad & Furman, Edward, 2012. "Pricing compound Poisson processes with the Farlie–Gumbel–Morgenstern dependence structure," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 151-157.
    13. Sharifah Farah Syed Yusoff Alhabshi & Zamira Hasanah Zamzuri & Siti Norafidah Mohd Ramli, 2021. "Monte Carlo Simulation of the Moments of a Copula-Dependent Risk Process with Weibull Interwaiting Time," Risks, MDPI, vol. 9(6), pages 1-21, June.
    14. Blier-Wong, Christopher & Cossette, Hélène & Marceau, Etienne, 2022. "Stochastic representation of FGM copulas using multivariate Bernoulli random variables," Computational Statistics & Data Analysis, Elsevier, vol. 173(C).
    15. Dominik Kortschak & Stéphane Loisel & Pierre Ribereau, 2014. "Ruin problems with worsening risks or with infinite mean claims," Post-Print hal-00735843, HAL.

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