April  2015, 11(2): 365-380. doi: 10.3934/jimo.2015.11.365

Credibility models with dependence structure over risks and time horizon

1. 

Department of Mathematics, Shanghai Maritime University, Shanghai, China

2. 

Center of International Finance and Risk Management, Department of Statistics and Actuarial Science, East China Normal University, Shanghai, China

Received  November 2012 Revised  March 2014 Published  September 2014

In this paper, the Bühlmann and Bühlmann-Straub's credibility models with a type of dependence structures over risks and over time are discussed. The inhomogeneous and homogeneous credibility estimators of risk premium were derived. The inhomogeneous credibility estimators for the existing credibility models with common effects are extended to slightly more general versions. The results obtained shake the classical meaning of the term ``credibility premiums''.
Citation: Weizhong Huang, Xianyi Wu. Credibility models with dependence structure over risks and time horizon. Journal of Industrial & Management Optimization, 2015, 11 (2) : 365-380. doi: 10.3934/jimo.2015.11.365
References:
[1]

C. Bolancé, M. Guillén, M. Denuit and J. Pinquet, Bonus-malus scales in segmented tariffs with stochastic migration between segments,, Insurance: Mathematics and Economics, 33 (2003), 273.  doi: 10.1016/S0167-6687(03)00139-2.  Google Scholar

[2]

H. Bühlmann, Experience rating and credibility,, Astin Bulletin, 4 (1967), 199.   Google Scholar

[3]

H. Bühlmann and E. Straub, Glaubwüdigkeit für Schadensäze,, Bulletin of the Swiss Association of Actuaries, 70 (1970), 111.   Google Scholar

[4]

H. Bühlmann and A. Gisler, A Course in Credibility Theory and its Applications,, Springer, (2005).   Google Scholar

[5]

D. R. Dannenburg, Crossed classification credibility models,, Transactions of the 25th International Congress of Actuaries, 4 (1995), 1.   Google Scholar

[6]

J. Dhaene, M. Denuit, M. J. Goovaerts, R. Kaas and D. Vyncke, The concept of comonotonicity in actuarial science and finance: Theory,, Insurance: Mathematics and Economics, 31 (2002), 3.  doi: 10.1016/S0167-6687(02)00134-8.  Google Scholar

[7]

J. Dhaene, M. Denuit, M. J. Goovaerts, R. Kaas and D. Vyncke, The concept of comonotonicity in actuarial science and finance: Applications,, Insurance: Mathematics and Economics, 31 (2002), 133.  doi: 10.1016/S0167-6687(02)00135-X.  Google Scholar

[8]

J. Dhaene and M. J. Goovaerts, Dependency of risks and stop-loss order,, Astin Bulletin, 26 (1996), 201.   Google Scholar

[9]

E. W. Frees, V. R. Young and Y. Luo, A Longitudinal Date Analysis Interpretation of Credibility models,, Insurance: Mathematics and Economics, 24 (1999), 229.  doi: 10.1016/S0167-6687(98)00055-9.  Google Scholar

[10]

E. W. Frees, V. R. Young and Y. Luo, Case studies using panel data models,, North American Actuarial Journal, 5 (2001), 24.  doi: 10.1080/10920277.2001.10596010.  Google Scholar

[11]

C. A. Hachemeister, Credibility for regression models with application to trend,, In Credibility, (1975), 129.   Google Scholar

[12]

W. S. Jewell, The use of collateral data in credibility theory: A hierarchical model,, Giorndle dell'lstituto Italianodegdi Attuari, 38 (1975), 1.   Google Scholar

[13]

T. Y. Lu and Y. Zhang, Generalized correlation order and stop-loss order,, Insurance: mathematics and economics, 35 (2004), 69.  doi: 10.1016/j.insmatheco.2004.04.003.  Google Scholar

[14]

A. Müller, Stop-loss order for portfolios of dependent risks,, Insurance: Mathematics and Economics, 21 (1997), 219.  doi: 10.1016/S0167-6687(97)00032-2.  Google Scholar

[15]

M. Pan, R. Wang and X. Wu, On the consistency of credibility premiums regarding Esscher principle,, Insurance: Mathematics and Economics, 42 (2008), 119.  doi: 10.1016/j.insmatheco.2007.01.009.  Google Scholar

[16]

O. Purcaru and M. Denuit, On the dependence induced by frequency credibility models,, Belgian Actuarial Bulletin, 2 (2002), 73.   Google Scholar

[17]

O. Purcaru and M. Denuit, Dependence in dynamic claim frequency credibility models,, Astin Bulletin, 33 (2003), 23.  doi: 10.2143/AST.33.1.1037.  Google Scholar

[18]

S. S. Wang, V. R. Young and H. H. Panjer, Axiomatic characterization of insurance prices,, Insurance: Mathematics and Economics, 21 (1997), 173.  doi: 10.1016/S0167-6687(97)00031-0.  Google Scholar

[19]

L. Wen, X. Wu and X. Zhao, The credibility premiums under generalized weighted loss functions,, Journal of Industrial and Management Optimization, 5 (2009), 893.  doi: 10.3934/jimo.2009.5.893.  Google Scholar

[20]

L. Wen, X. Wu and X. Zhou, The credibility premiums for models with dependence induced by common effects,, Insurance: Mathematics and Economics, 44 (2009), 19.  doi: 10.1016/j.insmatheco.2008.09.005.  Google Scholar

[21]

X. Wu and X. Zhou, A new characterization of distortion premiums via countable additivity for comonotonic risks,, Insurance: Mathematics and Economics, 38 (2006), 324.  doi: 10.1016/j.insmatheco.2005.09.002.  Google Scholar

[22]

K. L. Yeo and E. A. Valdez, Claim dependence with common effects in credibility models,, Insurance: Mathematics and Economics, 38 (2006), 609.  doi: 10.1016/j.insmatheco.2005.12.006.  Google Scholar

show all references

References:
[1]

C. Bolancé, M. Guillén, M. Denuit and J. Pinquet, Bonus-malus scales in segmented tariffs with stochastic migration between segments,, Insurance: Mathematics and Economics, 33 (2003), 273.  doi: 10.1016/S0167-6687(03)00139-2.  Google Scholar

[2]

H. Bühlmann, Experience rating and credibility,, Astin Bulletin, 4 (1967), 199.   Google Scholar

[3]

H. Bühlmann and E. Straub, Glaubwüdigkeit für Schadensäze,, Bulletin of the Swiss Association of Actuaries, 70 (1970), 111.   Google Scholar

[4]

H. Bühlmann and A. Gisler, A Course in Credibility Theory and its Applications,, Springer, (2005).   Google Scholar

[5]

D. R. Dannenburg, Crossed classification credibility models,, Transactions of the 25th International Congress of Actuaries, 4 (1995), 1.   Google Scholar

[6]

J. Dhaene, M. Denuit, M. J. Goovaerts, R. Kaas and D. Vyncke, The concept of comonotonicity in actuarial science and finance: Theory,, Insurance: Mathematics and Economics, 31 (2002), 3.  doi: 10.1016/S0167-6687(02)00134-8.  Google Scholar

[7]

J. Dhaene, M. Denuit, M. J. Goovaerts, R. Kaas and D. Vyncke, The concept of comonotonicity in actuarial science and finance: Applications,, Insurance: Mathematics and Economics, 31 (2002), 133.  doi: 10.1016/S0167-6687(02)00135-X.  Google Scholar

[8]

J. Dhaene and M. J. Goovaerts, Dependency of risks and stop-loss order,, Astin Bulletin, 26 (1996), 201.   Google Scholar

[9]

E. W. Frees, V. R. Young and Y. Luo, A Longitudinal Date Analysis Interpretation of Credibility models,, Insurance: Mathematics and Economics, 24 (1999), 229.  doi: 10.1016/S0167-6687(98)00055-9.  Google Scholar

[10]

E. W. Frees, V. R. Young and Y. Luo, Case studies using panel data models,, North American Actuarial Journal, 5 (2001), 24.  doi: 10.1080/10920277.2001.10596010.  Google Scholar

[11]

C. A. Hachemeister, Credibility for regression models with application to trend,, In Credibility, (1975), 129.   Google Scholar

[12]

W. S. Jewell, The use of collateral data in credibility theory: A hierarchical model,, Giorndle dell'lstituto Italianodegdi Attuari, 38 (1975), 1.   Google Scholar

[13]

T. Y. Lu and Y. Zhang, Generalized correlation order and stop-loss order,, Insurance: mathematics and economics, 35 (2004), 69.  doi: 10.1016/j.insmatheco.2004.04.003.  Google Scholar

[14]

A. Müller, Stop-loss order for portfolios of dependent risks,, Insurance: Mathematics and Economics, 21 (1997), 219.  doi: 10.1016/S0167-6687(97)00032-2.  Google Scholar

[15]

M. Pan, R. Wang and X. Wu, On the consistency of credibility premiums regarding Esscher principle,, Insurance: Mathematics and Economics, 42 (2008), 119.  doi: 10.1016/j.insmatheco.2007.01.009.  Google Scholar

[16]

O. Purcaru and M. Denuit, On the dependence induced by frequency credibility models,, Belgian Actuarial Bulletin, 2 (2002), 73.   Google Scholar

[17]

O. Purcaru and M. Denuit, Dependence in dynamic claim frequency credibility models,, Astin Bulletin, 33 (2003), 23.  doi: 10.2143/AST.33.1.1037.  Google Scholar

[18]

S. S. Wang, V. R. Young and H. H. Panjer, Axiomatic characterization of insurance prices,, Insurance: Mathematics and Economics, 21 (1997), 173.  doi: 10.1016/S0167-6687(97)00031-0.  Google Scholar

[19]

L. Wen, X. Wu and X. Zhao, The credibility premiums under generalized weighted loss functions,, Journal of Industrial and Management Optimization, 5 (2009), 893.  doi: 10.3934/jimo.2009.5.893.  Google Scholar

[20]

L. Wen, X. Wu and X. Zhou, The credibility premiums for models with dependence induced by common effects,, Insurance: Mathematics and Economics, 44 (2009), 19.  doi: 10.1016/j.insmatheco.2008.09.005.  Google Scholar

[21]

X. Wu and X. Zhou, A new characterization of distortion premiums via countable additivity for comonotonic risks,, Insurance: Mathematics and Economics, 38 (2006), 324.  doi: 10.1016/j.insmatheco.2005.09.002.  Google Scholar

[22]

K. L. Yeo and E. A. Valdez, Claim dependence with common effects in credibility models,, Insurance: Mathematics and Economics, 38 (2006), 609.  doi: 10.1016/j.insmatheco.2005.12.006.  Google Scholar

[1]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[2]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[3]

Xinpeng Wang, Bingo Wing-Kuen Ling, Wei-Chao Kuang, Zhijing Yang. Orthogonal intrinsic mode functions via optimization approach. Journal of Industrial & Management Optimization, 2021, 17 (1) : 51-66. doi: 10.3934/jimo.2019098

[4]

Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101

[5]

Kalikinkar Mandal, Guang Gong. On ideal $ t $-tuple distribution of orthogonal functions in filtering de bruijn generators. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020125

[6]

He Zhang, John Harlim, Xiantao Li. Estimating linear response statistics using orthogonal polynomials: An RKHS formulation. Foundations of Data Science, 2020, 2 (4) : 443-485. doi: 10.3934/fods.2020021

[7]

Hongwei Liu, Jingge Liu. On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020127

[8]

Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020032

[9]

Ténan Yeo. Stochastic and deterministic SIS patch model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021012

[10]

M. Dambrine, B. Puig, G. Vallet. A mathematical model for marine dinoflagellates blooms. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 615-633. doi: 10.3934/dcdss.2020424

[11]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[12]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464

[13]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366

[14]

Yu Jin, Xiang-Qiang Zhao. The spatial dynamics of a Zebra mussel model in river environments. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020362

[15]

Yunfeng Geng, Xiaoying Wang, Frithjof Lutscher. Coexistence of competing consumers on a single resource in a hybrid model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 269-297. doi: 10.3934/dcdsb.2020140

[16]

Pedro Aceves-Sanchez, Benjamin Aymard, Diane Peurichard, Pol Kennel, Anne Lorsignol, Franck Plouraboué, Louis Casteilla, Pierre Degond. A new model for the emergence of blood capillary networks. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2021001

[17]

Mingchao Zhao, You-Wei Wen, Michael Ng, Hongwei Li. A nonlocal low rank model for poisson noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021003

[18]

Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156

[19]

Jakub Kantner, Michal Beneš. Mathematical model of signal propagation in excitable media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 935-951. doi: 10.3934/dcdss.2020382

[20]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (40)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]