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October  2018, 14(4): 1545-1564. doi: 10.3934/jimo.2018020

Analysis of a dynamic premium strategy: From theoretical and marketing perspectives

Wing Yan Lee 1, and  Fangda Liu 2,,

1. 

Department of Mathematics and Statistics, Hang Seng Management College, Hang Shin Link, Siu Lek Yuen, Shatin, N.T., Hong Kong, China
2. 

China Institute for Actuarial Science, Central University of Finance and Economics, China

* Corresponding author: Fangda Liu

Received  February 2017 Revised  June 2017 Published  January 2018

Premium rate for an insurance policy is often reviewed and updated periodically according to past claim experience in real-life. In this paper, a dynamic premium strategy that depends on the past claim experience is proposed under the discrete-time risk model. The Gerber-Shiu function is analyzed under this model. The marketing implications of the dynamic premium strategy will also be discussed.

Keywords: Dynamic premium, discrete-time risk model, Gerber-Shiu function, geometric claims, marketing implications.
Mathematics Subject Classification: Primary: 60E99, 60D05.
Citation: Wing Yan Lee, Fangda Liu. Analysis of a dynamic premium strategy: From theoretical and marketing perspectives. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1545-1564. doi: 10.3934/jimo.2018020
References:
[1]

L. B. Afonso, A. D. Egidio dos Reis and H. R. Waters, Calculating continuous time ruin probabilities for a large portfolio with varying premium, ASTIN Bulletin, 39 (2009), 117-136.  doi: 10.2143/AST.39.1.2038059.  Google Scholar

[2]

L. B. Afonso, A. D. Egidio dos Reis and H. R. Waters, Numerical evaluation of continuous time ruin probabilities for a portfolio with credibility updated premiums, ASTIN Bulletin, 40 (2010), 399-414.  doi: 10.2143/AST.40.1.2049236.  Google Scholar

[3]

L. B. Afonso, Evaluation of ruin probabilities for surplus process with credibility and surplus dependent premium, Ph. D. Thesis, 2008. Google Scholar

[4]

S. Asmussen, On the ruin problem for some adapted premium rules, MaPhySto Research Report No. 5 University of Aarhus, Denmark., 1999. Google Scholar

[5] S. Asmussen and H. Albrecher, Ruin Probabilities, World Scientific, 2010.   Google Scholar
[6]

E. C. K. Cheung, D. Landriault, G. E. Willmot and J.-K. Woo, Gerber-Shiu analysis with a generalized penalty function, Scandinavian Actuarial Journal, 2010 (2010), 185-199.   Google Scholar

[7]

E. C. K. Cheung, D. Landriault, G. E. Willmot and J.-K. Woo, Structural properties of Gerber-Shiu function in dependent Sparre Andersen models, Insurance: Mathematics and Economics, 46 (2010), 117-126.  doi: 10.1016/j.insmatheco.2009.05.009.  Google Scholar

[8]

E. C. K. Cheung, D. Landriault, G. E. Willmot and J.-K. Woo, On orderings and bounds in a generalized sparre andersen risk model, Applied Stochastic Models in Business and Industry, 27 (2011), 51-60.  doi: 10.1002/asmb.837.  Google Scholar

[9]

H. U. Gerber and E. S. W. Shiu, On the time value of ruin, North American Actuarial Journal, 2 (1998), 48-78.  doi: 10.1080/10920277.1998.10595671.  Google Scholar

[10]

D. Landriault, C. Lemieux and G. E. Willmot, An adaptive premium policy with a Bayesian motivation in the classical risk model, Insurance: Mathematics and Economics, 51 (2012), 370-378.  doi: 10.1016/j.insmatheco.2012.06.001.  Google Scholar

[11]

S. Li, D. Landriault and C. Lemieux, A risk model with varying premiums: Its risk management implications, Insurance: Mathematics and Economics, 60 (2015), 38-46.  doi: 10.1016/j.insmatheco.2014.10.010.  Google Scholar

[12]

Z. Li and K. P. Sendova, On a ruin model with both interclaim times and premiums depending on claim sizes, Scandinavian Actuarial Journal, 2015 (2015), 245-265.   Google Scholar

[13]

S. Loisel and J. Trufin, Ultimate ruin probability in discrete time with Buhlmann credibility premium adjustments, Bulletin Francais d'Actuariat, 13 (2013), 73-102.   Google Scholar

[14]

C. C. -L. Tsai and G. Parker, Ruin probabilities: Classical versus credibility, NTU International Conference on Finance, 2004. Google Scholar

[15]

A. Tversky and D. Kahneman, Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty, 5 (1992), 297-323.   Google Scholar

[16]

J.-K. Woo, A generalized penalty function for a class of discrete renewal processes, Scandinavian Actuarial Journal, 2012 (2012), 130-152.   Google Scholar

[17]

X. Wu and S. Li, On the discounted penalty function in a discrete time renewal risk model with general interclaim times, Scandinavian Actuarial Journal, 2009 (2009), 281-294.   Google Scholar

[18]

Z. Zhang, Y. Yang and C. Liu, On a perturbed compound Poisson model with varying premium rates, Journal of Industrial and Management Optimization, 13 (2017), 721-736.   Google Scholar

show all references

References:
[1]

L. B. Afonso, A. D. Egidio dos Reis and H. R. Waters, Calculating continuous time ruin probabilities for a large portfolio with varying premium, ASTIN Bulletin, 39 (2009), 117-136.  doi: 10.2143/AST.39.1.2038059.  Google Scholar

[2]

L. B. Afonso, A. D. Egidio dos Reis and H. R. Waters, Numerical evaluation of continuous time ruin probabilities for a portfolio with credibility updated premiums, ASTIN Bulletin, 40 (2010), 399-414.  doi: 10.2143/AST.40.1.2049236.  Google Scholar

[3]

L. B. Afonso, Evaluation of ruin probabilities for surplus process with credibility and surplus dependent premium, Ph. D. Thesis, 2008. Google Scholar

[4]

S. Asmussen, On the ruin problem for some adapted premium rules, MaPhySto Research Report No. 5 University of Aarhus, Denmark., 1999. Google Scholar

[5] S. Asmussen and H. Albrecher, Ruin Probabilities, World Scientific, 2010.   Google Scholar
[6]

E. C. K. Cheung, D. Landriault, G. E. Willmot and J.-K. Woo, Gerber-Shiu analysis with a generalized penalty function, Scandinavian Actuarial Journal, 2010 (2010), 185-199.   Google Scholar

[7]

E. C. K. Cheung, D. Landriault, G. E. Willmot and J.-K. Woo, Structural properties of Gerber-Shiu function in dependent Sparre Andersen models, Insurance: Mathematics and Economics, 46 (2010), 117-126.  doi: 10.1016/j.insmatheco.2009.05.009.  Google Scholar

[8]

E. C. K. Cheung, D. Landriault, G. E. Willmot and J.-K. Woo, On orderings and bounds in a generalized sparre andersen risk model, Applied Stochastic Models in Business and Industry, 27 (2011), 51-60.  doi: 10.1002/asmb.837.  Google Scholar

[9]

H. U. Gerber and E. S. W. Shiu, On the time value of ruin, North American Actuarial Journal, 2 (1998), 48-78.  doi: 10.1080/10920277.1998.10595671.  Google Scholar

[10]

D. Landriault, C. Lemieux and G. E. Willmot, An adaptive premium policy with a Bayesian motivation in the classical risk model, Insurance: Mathematics and Economics, 51 (2012), 370-378.  doi: 10.1016/j.insmatheco.2012.06.001.  Google Scholar

[11]

S. Li, D. Landriault and C. Lemieux, A risk model with varying premiums: Its risk management implications, Insurance: Mathematics and Economics, 60 (2015), 38-46.  doi: 10.1016/j.insmatheco.2014.10.010.  Google Scholar

[12]

Z. Li and K. P. Sendova, On a ruin model with both interclaim times and premiums depending on claim sizes, Scandinavian Actuarial Journal, 2015 (2015), 245-265.   Google Scholar

[13]

S. Loisel and J. Trufin, Ultimate ruin probability in discrete time with Buhlmann credibility premium adjustments, Bulletin Francais d'Actuariat, 13 (2013), 73-102.   Google Scholar

[14]

C. C. -L. Tsai and G. Parker, Ruin probabilities: Classical versus credibility, NTU International Conference on Finance, 2004. Google Scholar

[15]

A. Tversky and D. Kahneman, Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty, 5 (1992), 297-323.   Google Scholar

[16]

J.-K. Woo, A generalized penalty function for a class of discrete renewal processes, Scandinavian Actuarial Journal, 2012 (2012), 130-152.   Google Scholar

[17]

X. Wu and S. Li, On the discounted penalty function in a discrete time renewal risk model with general interclaim times, Scandinavian Actuarial Journal, 2009 (2009), 281-294.   Google Scholar

[18]

Z. Zhang, Y. Yang and C. Liu, On a perturbed compound Poisson model with varying premium rates, Journal of Industrial and Management Optimization, 13 (2017), 721-736.   Google Scholar

Figure 1.  Strategy 1 ($c_{1} = 4$ and $c_{2} = 6$) vs Strategy 2 ($c = 4$) ($\eta_{1}$ denotes the starting premium)
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Figure 2.  Strategy 1 ($c_{1} = 4$ and $c_{2} = 6$) vs Strategy 2 ($c = 5$)
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Figure 3.  Strategy 1 ($c_{1} = 4$ and $c_{2} = 6$) vs Strategy 2 ($c = 4$)
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Figure 4.  Strategy 1 ($c_{1} = 4$ and $c_{2} = 6$) vs Strategy 2 ($c = 5$)
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