Improving Risk Assessment and Pricing with Dividend Barriers and Dependence Modelling: An Extension of the Cramer-Lundberg Model with Spearman Copulas

Kiswendsida Mahamoudou Ouedraogo; Delwendé Abdoul-Kabir Kafando; Frédéric Bere; Pierre Clovis Nitiema

doi:doi:10.11648/j.ijssam.20240901.11

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Improving Risk Assessment and Pricing with Dividend Barriers and Dependence Modelling: An Extension of the Cramer-Lundberg Model with Spearman Copulas

Kiswendsida Mahamoudou Ouedraogo, Delwendé Abdoul-Kabir Kafando^*, Frédéric Bere, Pierre Clovis Nitiema

Published in International Journal of Systems Science and Applied Mathematics (Volume 9, Issue 1)

Received: 7 December 2023 Accepted: 28 December 2023 Published: 11 January 2024

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Abstract

The compound Poisson risk model is a probabilistic model commonly used to evaluate the financial risk of an insurance company. This model assumes that claims arrive according to a Poisson process and that claim sizes follow an independent probability distribution. This paper presents an extension of this model, incorporating a dividend payment strategy with a constant threshold b. This extension allows for a better representation of the reality of insurance companies, which typically pay dividends to their shareholders. The traditional assumption of independence between claim sizes and interclaim intervals is also relaxed in this extension. This relaxation allows for recognition of the potential dependence between these variables, which can have a significant impact on the company’s ruin probability. The Spearman copula is used to model the dependent structure between claim sizes and interclaim intervals. The Spearman copula is a function that measures the degree of dependence between two variables. It is used in many fields, including insurance, finance, and statistics. The study focuses on the Laplace transform of the adjusted penalty function. The adjusted penalty function is a function that allows for the determination of the company’s ruin probability. The results of the study show that the dependence between claim sizes and interclaim intervals can have a significant impact on the company’s ruin probability. In particular, positive dependence between these variables can increase the ruin probability.

Published in	International Journal of Systems Science and Applied Mathematics (Volume 9, Issue 1)
DOI	10.11648/j.ijssam.20240901.11
Page(s)	1-8
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, Dependency, Integro-differential Equation, Laplace Transformation, Probability of Ruin

References

[1]	Cossette, H.; Marceau, E., F, (2014). On a compound poisson risk model with dependence and in a presence of a constant dividend barrier. Appl. Stoch. Models Bus. Ind., 30, 82-98.
[2]	S. Heilpern, (2014). Ruin measures for a compound Poisson risk model with dependence based on the Spearman copula and the exponential claim sizes. Insurance: Mathematics and economic, 59: 251-257.
[3]	Delwendé Abdoul-Kabir Kafando, Victorien Konané, Fréd éric Béré and Pierre Clovis Nitiéma, (2023). Extension of the Sparre Andersen via the Spearman copula, Advances and Applications in statistics. 86 (1), 79-100.
[4]	S. Asmussen, (1995). Stationary distributions for fluid flow models with or without Brownian noise, Communications in Statististics-stochastics Model. 11: 21-49.
[5]	Cosette, H., Marceau, E., and Marri, F., (2010). Analysis of ruin measure for the classical compound Poisson risk model with dependence. Scand. Actuar. J. 3, 221-245.
[6]	Nelsen, R. B., (2006). An introduction to copula, second edition: Springer Series in statistic, Springer-Verlag, New York.
[7]	H. Joe, (1997). Multivariate Models and Dependence Concepts, Chapman & Hall/CRC.
[8]	Hürlimann, W., (2004.a.). Multivariate Frechet copulas and conditional value- at- risque. Ind. J. Math. Sci. 7, 345-364.
[9]	M. Boudreault, (2003). “Modeling and pricing earthquake risk”, scor Canada Actuarial Price.
[10]	H. U. Gerber and E. S. W. Shiu, (1998). On the time value of ruin, North American Actuarial Journal, 48-78.
[11]	M.Boudreault, H.Cosette, D.LandriaultandE.Marceau, (2006). “On a risk model with dependence between interclaim arrivals and claim sizes”, Scandinavian Actuarial Journal, vol. 5, pp. 301-323.
[12]	A. K. Nikoloulopoulos and D. Karlis, (2008). “Fiiting copulas to bivariate earthquake data: the seismic gap hypothesis revisited”, Environmetrics, vol. 19, no. 3, PP. 251-269.
[13]	D Landriault, (2008). “Constant dividend barrier in a risk model with interclaim-dependent claim sizes”, Insurance: Mathematics and economics, vol. 42, no. 1, pp. 31-38.
[14]	K. C. Yue, G. Wang, and W. K. Li, (2007). “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.
[15]	H. U. Gerber, (1970). An extension of the renewal equation and its application in the collective theory of risk, Skandinavissk Actuarietidskrift. 205-210.
[16]	H. Albrecher, O. J. Boxma,(2004). A ruin model with dependance between claim sizes ans claim intervals, Insurance: Mathematics and Economics. 35: 245-254.
[17]	H. U. Gerber, E. S. W. Shiu, (2005). The time value of ruin in a sparre Andersen model, North American Actuarial Journal. 9 (2): 49-84.
[18]	Grandell, J.(1991)Aspectsofrisktheory. SpringerSeries in statistics: Probability and its applications. Springer- Verlag, New York.
[19]	Schmidt, V., T. Rolski, J. Teugels and H. Schmidli, (1991). Stochastic Processes for Insurance and Finance. Wiley.

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Ouedraogo, K. M., Kafando, D. A., Bere, F., Nitiema, P. C. (2024). Improving Risk Assessment and Pricing with Dividend Barriers and Dependence Modelling: An Extension of the Cramer-Lundberg Model with Spearman Copulas. International Journal of Systems Science and Applied Mathematics, 9(1), 1-8. https://doi.org/10.11648/j.ijssam.20240901.11

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ACS Style

Ouedraogo, K. M.; Kafando, D. A.; Bere, F.; Nitiema, P. C. Improving Risk Assessment and Pricing with Dividend Barriers and Dependence Modelling: An Extension of the Cramer-Lundberg Model with Spearman Copulas. Int. J. Syst. Sci. Appl. Math. 2024, 9(1), 1-8. doi: 10.11648/j.ijssam.20240901.11

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AMA Style

Ouedraogo KM, Kafando DA, Bere F, Nitiema PC. Improving Risk Assessment and Pricing with Dividend Barriers and Dependence Modelling: An Extension of the Cramer-Lundberg Model with Spearman Copulas. Int J Syst Sci Appl Math. 2024;9(1):1-8. doi: 10.11648/j.ijssam.20240901.11

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@article{10.11648/j.ijssam.20240901.11,
  author = {Kiswendsida Mahamoudou Ouedraogo and Delwendé Abdoul-Kabir Kafando and Frédéric Bere and Pierre Clovis Nitiema},
  title = {Improving Risk Assessment and Pricing with Dividend Barriers and Dependence Modelling: An Extension of the Cramer-Lundberg Model with Spearman Copulas},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {9},
  number = {1},
  pages = {1-8},
  doi = {10.11648/j.ijssam.20240901.11},
  url = {https://doi.org/10.11648/j.ijssam.20240901.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20240901.11},
  abstract = {The compound Poisson risk model is a probabilistic model commonly used to evaluate the financial risk of an insurance company. This model assumes that claims arrive according to a Poisson process and that claim sizes follow an independent probability distribution. This paper presents an extension of this model, incorporating a dividend payment strategy with a constant threshold b. This extension allows for a better representation of the reality of insurance companies, which typically pay dividends to their shareholders. The traditional assumption of independence between claim sizes and interclaim intervals is also relaxed in this extension. This relaxation allows for recognition of the potential dependence between these variables, which can have a significant impact on the company’s ruin probability. The Spearman copula is used to model the dependent structure between claim sizes and interclaim intervals. The Spearman copula is a function that measures the degree of dependence between two variables. It is used in many fields, including insurance, finance, and statistics. The study focuses on the Laplace transform of the adjusted penalty function. The adjusted penalty function is a function that allows for the determination of the company’s ruin probability. The results of the study show that the dependence between claim sizes and interclaim intervals can have a significant impact on the company’s ruin probability. In particular, positive dependence between these variables can increase the ruin probability.},
 year = {2024}
}

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TY  - JOUR
T1  - Improving Risk Assessment and Pricing with Dividend Barriers and Dependence Modelling: An Extension of the Cramer-Lundberg Model with Spearman Copulas
AU  - Kiswendsida Mahamoudou Ouedraogo
AU  - Delwendé Abdoul-Kabir Kafando
AU  - Frédéric Bere
AU  - Pierre Clovis Nitiema
Y1  - 2024/01/11
PY  - 2024
N1  - https://doi.org/10.11648/j.ijssam.20240901.11
DO  - 10.11648/j.ijssam.20240901.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
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PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20240901.11
AB  - The compound Poisson risk model is a probabilistic model commonly used to evaluate the financial risk of an insurance company. This model assumes that claims arrive according to a Poisson process and that claim sizes follow an independent probability distribution. This paper presents an extension of this model, incorporating a dividend payment strategy with a constant threshold b. This extension allows for a better representation of the reality of insurance companies, which typically pay dividends to their shareholders. The traditional assumption of independence between claim sizes and interclaim intervals is also relaxed in this extension. This relaxation allows for recognition of the potential dependence between these variables, which can have a significant impact on the company’s ruin probability. The Spearman copula is used to model the dependent structure between claim sizes and interclaim intervals. The Spearman copula is a function that measures the degree of dependence between two variables. It is used in many fields, including insurance, finance, and statistics. The study focuses on the Laplace transform of the adjusted penalty function. The adjusted penalty function is a function that allows for the determination of the company’s ruin probability. The results of the study show that the dependence between claim sizes and interclaim intervals can have a significant impact on the company’s ruin probability. In particular, positive dependence between these variables can increase the ruin probability.
VL  - 9
IS  - 1
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Author Information

Kiswendsida Mahamoudou Ouedraogo

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

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Delwendé Abdoul-Kabir Kafando

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

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Frédéric Bere

Department of Mathematics,Ecole Normale Supérieure, Ouagadougou, Burkina Faso

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Pierre Clovis Nitiema

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

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Plain Text BibTeX RIS

APA Style

Ouedraogo, K. M., Kafando, D. A., Bere, F., Nitiema, P. C. (2024). Improving Risk Assessment and Pricing with Dividend Barriers and Dependence Modelling: An Extension of the Cramer-Lundberg Model with Spearman Copulas. International Journal of Systems Science and Applied Mathematics, 9(1), 1-8. https://doi.org/10.11648/j.ijssam.20240901.11

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ACS Style

Ouedraogo, K. M.; Kafando, D. A.; Bere, F.; Nitiema, P. C. Improving Risk Assessment and Pricing with Dividend Barriers and Dependence Modelling: An Extension of the Cramer-Lundberg Model with Spearman Copulas. Int. J. Syst. Sci. Appl. Math. 2024, 9(1), 1-8. doi: 10.11648/j.ijssam.20240901.11

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AMA Style

Ouedraogo KM, Kafando DA, Bere F, Nitiema PC. Improving Risk Assessment and Pricing with Dividend Barriers and Dependence Modelling: An Extension of the Cramer-Lundberg Model with Spearman Copulas. Int J Syst Sci Appl Math. 2024;9(1):1-8. doi: 10.11648/j.ijssam.20240901.11

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@article{10.11648/j.ijssam.20240901.11,
  author = {Kiswendsida Mahamoudou Ouedraogo and Delwendé Abdoul-Kabir Kafando and Frédéric Bere and Pierre Clovis Nitiema},
  title = {Improving Risk Assessment and Pricing with Dividend Barriers and Dependence Modelling: An Extension of the Cramer-Lundberg Model with Spearman Copulas},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {9},
  number = {1},
  pages = {1-8},
  doi = {10.11648/j.ijssam.20240901.11},
  url = {https://doi.org/10.11648/j.ijssam.20240901.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20240901.11},
  abstract = {The compound Poisson risk model is a probabilistic model commonly used to evaluate the financial risk of an insurance company. This model assumes that claims arrive according to a Poisson process and that claim sizes follow an independent probability distribution. This paper presents an extension of this model, incorporating a dividend payment strategy with a constant threshold b. This extension allows for a better representation of the reality of insurance companies, which typically pay dividends to their shareholders. The traditional assumption of independence between claim sizes and interclaim intervals is also relaxed in this extension. This relaxation allows for recognition of the potential dependence between these variables, which can have a significant impact on the company’s ruin probability. The Spearman copula is used to model the dependent structure between claim sizes and interclaim intervals. The Spearman copula is a function that measures the degree of dependence between two variables. It is used in many fields, including insurance, finance, and statistics. The study focuses on the Laplace transform of the adjusted penalty function. The adjusted penalty function is a function that allows for the determination of the company’s ruin probability. The results of the study show that the dependence between claim sizes and interclaim intervals can have a significant impact on the company’s ruin probability. In particular, positive dependence between these variables can increase the ruin probability.},
 year = {2024}
}

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TY  - JOUR
T1  - Improving Risk Assessment and Pricing with Dividend Barriers and Dependence Modelling: An Extension of the Cramer-Lundberg Model with Spearman Copulas
AU  - Kiswendsida Mahamoudou Ouedraogo
AU  - Delwendé Abdoul-Kabir Kafando
AU  - Frédéric Bere
AU  - Pierre Clovis Nitiema
Y1  - 2024/01/11
PY  - 2024
N1  - https://doi.org/10.11648/j.ijssam.20240901.11
DO  - 10.11648/j.ijssam.20240901.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 1
EP  - 8
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20240901.11
AB  - The compound Poisson risk model is a probabilistic model commonly used to evaluate the financial risk of an insurance company. This model assumes that claims arrive according to a Poisson process and that claim sizes follow an independent probability distribution. This paper presents an extension of this model, incorporating a dividend payment strategy with a constant threshold b. This extension allows for a better representation of the reality of insurance companies, which typically pay dividends to their shareholders. The traditional assumption of independence between claim sizes and interclaim intervals is also relaxed in this extension. This relaxation allows for recognition of the potential dependence between these variables, which can have a significant impact on the company’s ruin probability. The Spearman copula is used to model the dependent structure between claim sizes and interclaim intervals. The Spearman copula is a function that measures the degree of dependence between two variables. It is used in many fields, including insurance, finance, and statistics. The study focuses on the Laplace transform of the adjusted penalty function. The adjusted penalty function is a function that allows for the determination of the company’s ruin probability. The results of the study show that the dependence between claim sizes and interclaim intervals can have a significant impact on the company’s ruin probability. In particular, positive dependence between these variables can increase the ruin probability.
VL  - 9
IS  - 1
ER  -

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