On Sparre Andersen Model with Partial Premium Payment Strategy to Shareholders with Dependence via Sperman Copula

Delwendé Abdoul-Kabir Kafando; François Xavier Ouedraogo; Lassané Sawadogo; Kiswendsida Mahamoudou Ouedraogo; Pierre Clovis Nitiema

doi:doi:10.11648/j.ajtas.20241301.11

Research Article |

On Sparre Andersen Model with Partial Premium Payment Strategy to Shareholders with Dependence via Sperman Copula

Delwendé Abdoul-Kabir Kafando, François Xavier Ouedraogo, Lassané Sawadogo, Kiswendsida Mahamoudou Ouedraogo, Pierre Clovis Nitiema

Published in American Journal of Theoretical and Applied Statistics (Volume 13, Issue 1)

Received: 8 December 2023 Accepted: 22 December 2023 Published: 8 January 2024

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Abstract

This paper is based on the Poisson composite risk model, popularised for its flexibility in modelling loss occurrences. However, it innovates by incorporating a strategy of distributing dividends to shareholders, adding a realistic dimension to the financial implications. A key element is the introduction of a constant threshold 'b', representing a critical amount beyond which claims become significant. This threshold makes it possible to distinguish between small, routine claims and major events with a significant impact on reserves. In addition, the model introduces a dependency between the amount of claims and the time between claims via the Spearman copula. This copula captures the non-independence often observed in insurance data, where large claims tend to be followed by claim-free periods or vice versa. The analysis then focuses on the integro-differential equation associated with the model, which describes the evolution of Gerber's Shiu function, a fundamental element in assessing the reserve required to cover future obligations. The Laplace transform of this function is also studied, providing valuable information on the distribution of the long-term reserve.

Published in	American Journal of Theoretical and Applied Statistics (Volume 13, Issue 1)
DOI	10.11648/j.ajtas.20241301.11
Page(s)	1-7
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Gerber-Shiu Functions, Dependence, Spearman Copula, Dividends, Integro-Differential Equation

References

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[3]	Delwendé Abdoul-Kabir Kafando, Victorien Konané, Frédéric Béré and Pierre Clovis Nitiéma, Extension of the Sparre Andersen via the Spearman copula, Advances and Applications in statistics 86(1) (2023), 79-100.
[4]	Kiswendsida Mahamoudou OUEDRAOGO, Francois Xavier OUEDRAOGO, Delwendé Abdoul-Kabir KAFANDO and Pierre Clovis NITIEMA. On compound risk model with partial premium payment strategy to shareholders and dependence between claim amount and inter-claim times through the Spearman copula, Advances and applications in statistics 89(2)(2023), 175-188.
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[12]	M. Boudreault, H. Cosette, D. Landriault and E. Marceau, «On a risk model with dependence between interclaim arrivals and claim sizes,» Scandinavian Actuarial Journal, vol. 5, pp. 301-323, 2006.
[13]	A. K. Nikoloulopoulos and D. Karlis, «Fiiting copulas to bivariate earthquake data: the seismic gap hypothesis revisited,» Environmetrics, vol.19, no. 3, PP. 251-269, 2008.
[14]	D Landriault, «Constant dividend barrier in a risk model with interclaim-dependent claim sizes,» Insurance: Mathematics and economics, vol. 42, no. 1, pp. 31-38, 2008.
[15]	K. C. Yue, G. Wang, and W. K. Li, «The Gerber Shiu expected discounted penalty function for risk process with interest and a constant dividend barrier,» Insurance: Mathematics and economics, vol. 40, no.1, pp. 104-112, 2007.
[16]	X. S. Lin, G. E. Wilmot, and S. Drekic, «The classical risk model with a constant dividend barrier: analysis of the Gerber Shiu discounted penalty function,» Insurance: Mathematics and economics, vol. 33, no. 3, pp. 551-556, 2003.
[17]	H. U. Gerber, An extension of the renewal equation and its application in the collective theory of risk, Skandinavissk Actuarietidskrift (1970) 205-210.
[18]	Delwendé Abdoul-Kabir KAFANDO, Frédéric BÉRÉ, Victorien KONANÉ and Pierre Clovis NITIÉMA, Extension of the compound Poisson model via the Spearman copula, Far East Journal of Theoretical Statistics 67(2) (2023), 147-184. http://dx.doi.org/10.17654/0972086323008
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APA Style

Kafando, D. A., Ouedraogo, F. X., Sawadogo, L., Ouedraogo, K. M., Nitiema, P. C. (2024). On Sparre Andersen Model with Partial Premium Payment Strategy to Shareholders with Dependence via Sperman Copula. American Journal of Theoretical and Applied Statistics, 13(1), 1-7. https://doi.org/10.11648/j.ajtas.20241301.11

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ACS Style

Kafando, D. A.; Ouedraogo, F. X.; Sawadogo, L.; Ouedraogo, K. M.; Nitiema, P. C. On Sparre Andersen Model with Partial Premium Payment Strategy to Shareholders with Dependence via Sperman Copula. Am. J. Theor. Appl. Stat. 2024, 13(1), 1-7. doi: 10.11648/j.ajtas.20241301.11

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AMA Style

Kafando DA, Ouedraogo FX, Sawadogo L, Ouedraogo KM, Nitiema PC. On Sparre Andersen Model with Partial Premium Payment Strategy to Shareholders with Dependence via Sperman Copula. Am J Theor Appl Stat. 2024;13(1):1-7. doi: 10.11648/j.ajtas.20241301.11

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@article{10.11648/j.ajtas.20241301.11,
  author = {Delwendé Abdoul-Kabir Kafando and François Xavier Ouedraogo and Lassané Sawadogo and Kiswendsida Mahamoudou Ouedraogo and Pierre Clovis Nitiema},
  title = {On Sparre Andersen Model with Partial Premium Payment Strategy to Shareholders with Dependence via Sperman Copula},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {13},
  number = {1},
  pages = {1-7},
  doi = {10.11648/j.ajtas.20241301.11},
  url = {https://doi.org/10.11648/j.ajtas.20241301.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20241301.11},
  abstract = {This paper is based on the Poisson composite risk model, popularised for its flexibility in modelling loss occurrences. However, it innovates by incorporating a strategy of distributing dividends to shareholders, adding a realistic dimension to the financial implications. A key element is the introduction of a constant threshold 'b', representing a critical amount beyond which claims become significant. This threshold makes it possible to distinguish between small, routine claims and major events with a significant impact on reserves. In addition, the model introduces a dependency between the amount of claims and the time between claims via the Spearman copula. This copula captures the non-independence often observed in insurance data, where large claims tend to be followed by claim-free periods or vice versa. The analysis then focuses on the integro-differential equation associated with the model, which describes the evolution of Gerber's Shiu function, a fundamental element in assessing the reserve required to cover future obligations. The Laplace transform of this function is also studied, providing valuable information on the distribution of the long-term reserve.
},
 year = {2024}
}

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TY  - JOUR
T1  - On Sparre Andersen Model with Partial Premium Payment Strategy to Shareholders with Dependence via Sperman Copula
AU  - Delwendé Abdoul-Kabir Kafando
AU  - François Xavier Ouedraogo
AU  - Lassané Sawadogo
AU  - Kiswendsida Mahamoudou Ouedraogo
AU  - Pierre Clovis Nitiema
Y1  - 2024/01/08
PY  - 2024
N1  - https://doi.org/10.11648/j.ajtas.20241301.11
DO  - 10.11648/j.ajtas.20241301.11
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 1
EP  - 7
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20241301.11
AB  - This paper is based on the Poisson composite risk model, popularised for its flexibility in modelling loss occurrences. However, it innovates by incorporating a strategy of distributing dividends to shareholders, adding a realistic dimension to the financial implications. A key element is the introduction of a constant threshold 'b', representing a critical amount beyond which claims become significant. This threshold makes it possible to distinguish between small, routine claims and major events with a significant impact on reserves. In addition, the model introduces a dependency between the amount of claims and the time between claims via the Spearman copula. This copula captures the non-independence often observed in insurance data, where large claims tend to be followed by claim-free periods or vice versa. The analysis then focuses on the integro-differential equation associated with the model, which describes the evolution of Gerber's Shiu function, a fundamental element in assessing the reserve required to cover future obligations. The Laplace transform of this function is also studied, providing valuable information on the distribution of the long-term reserve.

VL  - 13
IS  - 1
ER  -

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Author Information

Delwendé Abdoul-Kabir Kafando

Department of Mathematics, Université Joseph KI-ZERBO, Ouagadougou, Burkina Faso

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François Xavier Ouedraogo

Department of Mathematics, Université Joseph KI-ZERBO, Ouagadougou, Burkina Faso

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Lassané Sawadogo

Department of Mathematics, Université Joseph KI-ZERBO, Ouagadougou, Burkina Faso
Kiswendsida Mahamoudou Ouedraogo

Department of Mathematics, Université Joseph KI-ZERBO, Ouagadougou, Burkina Faso

Contact Email
Pierre Clovis Nitiema

Department of Mathematics, Université Thomas SANKARA, Ouagadougou, Burkina Faso

Contact Email

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Plain Text BibTeX RIS

APA Style

Kafando, D. A., Ouedraogo, F. X., Sawadogo, L., Ouedraogo, K. M., Nitiema, P. C. (2024). On Sparre Andersen Model with Partial Premium Payment Strategy to Shareholders with Dependence via Sperman Copula. American Journal of Theoretical and Applied Statistics, 13(1), 1-7. https://doi.org/10.11648/j.ajtas.20241301.11

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ACS Style

Kafando, D. A.; Ouedraogo, F. X.; Sawadogo, L.; Ouedraogo, K. M.; Nitiema, P. C. On Sparre Andersen Model with Partial Premium Payment Strategy to Shareholders with Dependence via Sperman Copula. Am. J. Theor. Appl. Stat. 2024, 13(1), 1-7. doi: 10.11648/j.ajtas.20241301.11

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AMA Style

Kafando DA, Ouedraogo FX, Sawadogo L, Ouedraogo KM, Nitiema PC. On Sparre Andersen Model with Partial Premium Payment Strategy to Shareholders with Dependence via Sperman Copula. Am J Theor Appl Stat. 2024;13(1):1-7. doi: 10.11648/j.ajtas.20241301.11

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@article{10.11648/j.ajtas.20241301.11,
  author = {Delwendé Abdoul-Kabir Kafando and François Xavier Ouedraogo and Lassané Sawadogo and Kiswendsida Mahamoudou Ouedraogo and Pierre Clovis Nitiema},
  title = {On Sparre Andersen Model with Partial Premium Payment Strategy to Shareholders with Dependence via Sperman Copula},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {13},
  number = {1},
  pages = {1-7},
  doi = {10.11648/j.ajtas.20241301.11},
  url = {https://doi.org/10.11648/j.ajtas.20241301.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20241301.11},
  abstract = {This paper is based on the Poisson composite risk model, popularised for its flexibility in modelling loss occurrences. However, it innovates by incorporating a strategy of distributing dividends to shareholders, adding a realistic dimension to the financial implications. A key element is the introduction of a constant threshold 'b', representing a critical amount beyond which claims become significant. This threshold makes it possible to distinguish between small, routine claims and major events with a significant impact on reserves. In addition, the model introduces a dependency between the amount of claims and the time between claims via the Spearman copula. This copula captures the non-independence often observed in insurance data, where large claims tend to be followed by claim-free periods or vice versa. The analysis then focuses on the integro-differential equation associated with the model, which describes the evolution of Gerber's Shiu function, a fundamental element in assessing the reserve required to cover future obligations. The Laplace transform of this function is also studied, providing valuable information on the distribution of the long-term reserve.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - On Sparre Andersen Model with Partial Premium Payment Strategy to Shareholders with Dependence via Sperman Copula
AU  - Delwendé Abdoul-Kabir Kafando
AU  - François Xavier Ouedraogo
AU  - Lassané Sawadogo
AU  - Kiswendsida Mahamoudou Ouedraogo
AU  - Pierre Clovis Nitiema
Y1  - 2024/01/08
PY  - 2024
N1  - https://doi.org/10.11648/j.ajtas.20241301.11
DO  - 10.11648/j.ajtas.20241301.11
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 1
EP  - 7
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20241301.11
AB  - This paper is based on the Poisson composite risk model, popularised for its flexibility in modelling loss occurrences. However, it innovates by incorporating a strategy of distributing dividends to shareholders, adding a realistic dimension to the financial implications. A key element is the introduction of a constant threshold 'b', representing a critical amount beyond which claims become significant. This threshold makes it possible to distinguish between small, routine claims and major events with a significant impact on reserves. In addition, the model introduces a dependency between the amount of claims and the time between claims via the Spearman copula. This copula captures the non-independence often observed in insurance data, where large claims tend to be followed by claim-free periods or vice versa. The analysis then focuses on the integro-differential equation associated with the model, which describes the evolution of Gerber's Shiu function, a fundamental element in assessing the reserve required to cover future obligations. The Laplace transform of this function is also studied, providing valuable information on the distribution of the long-term reserve.

VL  - 13
IS  - 1
ER  -

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