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

Credibility models with dependence structure over risks and time horizon

Weizhong Huang 1, and  Xianyi Wu 2,

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''.
Keywords: orthogonal projection, credibility factor, dependence., Credibility model.
Mathematics Subject Classification: Primary: 91B30, 62P05; Secondary: 97M3.
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

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