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Discrete-Time Risk Models with Claim Correlated Premiums in a Markovian Environment

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  • Dhiti Osatakul

    (Department of Economics, The University of Melbourne, Parkville, VIC 3010, Australia
    These authors contributed equally to this work.)

  • Xueyuan Wu

    (Department of Economics, The University of Melbourne, Parkville, VIC 3010, Australia
    These authors contributed equally to this work.)

Abstract

In this paper we consider a discrete-time risk model, which allows the premium to be adjusted according to claims experience. This model is inspired by the well-known bonus-malus system in the non-life insurance industry. Two strategies of adjusting periodic premiums are considered: aggregate claims or claim frequency. Recursive formulae are derived to compute the finite-time ruin probabilities, and Lundberg-type upper bounds are also derived to evaluate the ultimate-time ruin probabilities. In addition, we extend the risk model by considering an external Markovian environment in which the claims distributions are governed by an external Markov process so that the periodic premium adjustments vary when the external environment state changes. We then study the joint distribution of premium level and environment state at ruin given ruin occurs. Two numerical examples are provided at the end of this paper to illustrate the impact of the initial external environment state, the initial premium level and the initial surplus on the ruin probability.

Suggested Citation

  • Dhiti Osatakul & Xueyuan Wu, 2021. "Discrete-Time Risk Models with Claim Correlated Premiums in a Markovian Environment," Risks, MDPI, vol. 9(1), pages 1-23, January.
    • Handle: RePEc:gam:jrisks:v:9:y:2021:i:1:p:26-:d:479949
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    References listed on IDEAS

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    1. Cai, Jun & Dickson, David C.M., 2004. "Ruin probabilities with a Markov chain interest model," Insurance: Mathematics and Economics, Elsevier, vol. 35(3), pages 513-525, December.
    2. Ágoston, Kolos Csaba & Gyetvai, Márton, 2020. "Joint Optimization Of Transition Rules And The Premium Scale In A Bonus-Malus System," ASTIN Bulletin, Cambridge University Press, vol. 50(3), pages 743-776, September.
    3. Afonso, Lourdes B. & Cardoso, Rui M. R. & Egídio dos Reis, Alfredo D. & Guerreiro, Gracinda Rita, 2017. "Measuring The Impact Of A Bonus-Malus System In Finite And Continuous Time Ruin Probabilities For Large Portfolios In Motor Insurance," ASTIN Bulletin, Cambridge University Press, vol. 47(2), pages 417-435, May.
    4. Georges Dionne & Olfa Ghali, 2005. "The (1992) Bonus‐Malus System in Tunisia: An Empirical Evaluation," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 72(4), pages 609-633, December.
    5. Corina Constantinescu & Suhang Dai & Weihong Ni & Zbigniew Palmowski, 2016. "Ruin Probabilities with Dependence on the Number of Claims within a Fixed Time Window," Risks, MDPI, vol. 4(2), pages 1-23, June.
    6. Pinquet, Jean & Guillén, Montserrat & Bolancé, Catalina, 2001. "Allowance for the Age of Claims in Bonus-Malus Systems," ASTIN Bulletin, Cambridge University Press, vol. 31(2), pages 337-348, November.
    7. Denuit, Michel & Guillen, Montserrat & Trufin, Julien, 2019. "Multivariate credibility modelling for usage-based motor insurance pricing with behavioural data," LIDAM Reprints ISBA 2019039, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    8. Lourdes B. Afonso & Rui M. R. Cardoso & Alfredo D. Egídio dos Reis & Gracinda R. Guerreiro, 2020. "Ruin Probabilities And Capital Requirement for Open Automobile Portfolios With a Bonus‐Malus System Based on Claim Counts," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 87(2), pages 501-522, June.
    9. Li, Bo & Ni, Weihong & Constantinescu, Corina, 2015. "Risk models with premiums adjusted to claims number," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 94-102.
    10. Afonso, Lourdes B. & dos Reis, Alfredo D. Egídio & Waters, Howard R., 2009. "Calculating Continuous Time Ruin Probabilities for a Large Portfolio with Varying Premiums," ASTIN Bulletin, Cambridge University Press, vol. 39(1), pages 117-136, May.
    11. Denuit, Michel & Guillen, Montserrat & Trufin, Julien, 2019. "Multivariate credibility modelling for usage-based motor insurance pricing with behavioural data," Annals of Actuarial Science, Cambridge University Press, vol. 13(2), pages 378-399, September.
    12. Niemiec, Małgorzata, 2007. "Bonus-malus Systems as Markov Set-chains," ASTIN Bulletin, Cambridge University Press, vol. 37(1), pages 53-65, May.
    13. Bong-Joo Lee & Dae-Hwan Kim, 2016. "Moral Hazard in Insurance Claiming from a Korean Natural Experiment," The Geneva Papers on Risk and Insurance - Issues and Practice, Palgrave Macmillan;The Geneva Association, vol. 41(3), pages 455-467, July.
    14. De Pril, Nelson, 1979. "Optimal Claim Decisions for a Bonus-Malus System: a Continuous Approach," ASTIN Bulletin, Cambridge University Press, vol. 10(2), pages 215-222, March.
    15. Viswanathan, Krupa S. & Lemaire, Jean, 2005. "Bonus-malus Systems in a Deregulated Environment: Forecasting Market Shares Using Diffusion Models," ASTIN Bulletin, Cambridge University Press, vol. 35(1), pages 299-319, May.
    16. Baione, Fabio & Levantesi, Susanna & Menzietti, Massimiliano, 2002. "The Development of an Optimal Bonus-Malus System in a Competitive Market," ASTIN Bulletin, Cambridge University Press, vol. 32(1), pages 159-170, May.
    17. Lemaire, Jean & Zi, Hongmin, 1994. "A Comparative Analysis of 30 Bonus-Malus Systems," ASTIN Bulletin, Cambridge University Press, vol. 24(2), pages 287-309, November.
    18. Wagner, Christian, 2002. "Time in the red in a two state Markov model," Insurance: Mathematics and Economics, Elsevier, vol. 31(3), pages 365-372, December.
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    Cited by:

    1. Osatakul, Dhiti & Li, Shuanming & Wu, Xueyuan, 2023. "Discrete-time risk models with surplus-dependent premium corrections," Applied Mathematics and Computation, Elsevier, vol. 437(C).
    2. Jingchao Li & Bihao Su & Zhenghong Wei & Ciyu Nie, 2022. "A Multinomial Approximation Approach for the Finite Time Survival Probability Under the Markov-modulated Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 2169-2194, September.

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