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A multivariate aggregate loss model

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  • Ren, Jiandong

Abstract

In this paper, we introduce a multivariate aggregate loss model, where multiple categories of losses are considered. The model assumes that different types of claims arrive according to a Marked Markovian arrival process (MMAP) introduced by He and Neuts (1998) in the queuing literature. This approach enables us to allow dependencies among the claim frequencies, and among the claim sizes, as well as between claim frequencies and claim sizes. This model extends the (univariate) Markov modulated risk processes (sometimes referred to as regime switching models) widely used in insurance and financial analysis. For the proposed model, we provide formulas for calculating the joint moments of the present value of aggregate claims occurring in any time interval (0,t]. Numerical examples are provided to show possible applications of the model.

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  • Ren, Jiandong, 2012. "A multivariate aggregate loss model," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 402-408.
    • Handle: RePEc:eee:insuma:v:51:y:2012:i:2:p:402-408
    DOI: 10.1016/j.insmatheco.2012.06.009
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    References listed on IDEAS

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    1. Mary Hardy, 2001. "A Regime-Switching Model of Long-Term Stock Returns," North American Actuarial Journal, Taylor & Francis Journals, vol. 5(2), pages 41-53.
    2. Frees, Edward W. & Valdez, Emiliano A., 2008. "Hierarchical Insurance Claims Modeling," Journal of the American Statistical Association, American Statistical Association, vol. 103(484), pages 1457-1469.
    3. He, Qi-Ming & Neuts, Marcel F., 1998. "Markov chains with marked transitions," Stochastic Processes and their Applications, Elsevier, vol. 74(1), pages 37-52, May.
    4. Kim, Bara & Kim, Hwa-Sung, 2007. "Moments of claims in a Markovian environment," Insurance: Mathematics and Economics, Elsevier, vol. 40(3), pages 485-497, May.
    5. Frees, Edward W. (Jed) & Meyers, Glenn & Cummings, A. David, 2010. "Dependent Multi-Peril Ratemaking Models," ASTIN Bulletin, Cambridge University Press, vol. 40(2), pages 699-726, November.
    6. Odd O. Aalen & Ørnulf Borgan & Harald Fekjær, 2001. "Covariate Adjustment of Event Histories Estimated from Markov Chains: The Additive Approach," Biometrics, The International Biometric Society, vol. 57(4), pages 993-1001, December.
    7. Sundt, Bjørn, 1999. "On Multivariate Panjer Recursions," ASTIN Bulletin, Cambridge University Press, vol. 29(1), pages 29-45, May.
    8. Delbaen, F. & Haezendonck, J., 1987. "Classical risk theory in an economic environment," Insurance: Mathematics and Economics, Elsevier, vol. 6(2), pages 85-116, April.
    9. Taylor, G. C., 1979. "Probability of Ruin under Inflationary Conditions or under Experience Rating," ASTIN Bulletin, Cambridge University Press, vol. 10(2), pages 149-162, March.
    10. Leveille, Ghislain & Garrido, Jose, 2001. "Moments of compound renewal sums with discounted claims," Insurance: Mathematics and Economics, Elsevier, vol. 28(2), pages 217-231, April.
    11. Albrecher, Hansjorg & Boxma, Onno J., 2005. "On the discounted penalty function in a Markov-dependent risk model," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 650-672, December.
    12. Hesselager, Ole, 1996. "Recursions for certain bivariate counting distributions and their compound distributions," ASTIN Bulletin, Cambridge University Press, vol. 26(1), pages 35-52, May.
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    Cited by:

    1. Ramírez-Cobo, Pepa & Carrizosa, Emilio & Lillo, Rosa E., 2021. "Analysis of an aggregate loss model in a Markov renewal regime," Applied Mathematics and Computation, Elsevier, vol. 396(C).
    2. Hyunjoo Yoo & Bara Kim & Jeongsim Kim & Jiwook Jang, 2020. "Transform approach for discounted aggregate claims in a risk model with descendant claims," Annals of Operations Research, Springer, vol. 293(1), pages 175-192, October.
    3. Guillén, Montserrat & Sarabia, José María & Prieto, Faustino, 2013. "Simple risk measure calculations for sums of positive random variables," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 273-280.
    4. Övgücan Karadağ Erdemir, 2023. "A Comparative Perspective on Multivariate Modeling of Insurance Compensation Payments with Regression-Based and Copula-Based Models," EKOIST Journal of Econometrics and Statistics, Istanbul University, Faculty of Economics, vol. 0(39), pages 161-171, December.

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