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Neumann Series on the Recursive Moments of Copula-Dependent Aggregate Discounted Claims

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  • Siti Norafidah Mohd Ramli

    (Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney 2109, Australia)

  • Jiwook Jang

    (Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney 2109, Australia)

Abstract

We study the recursive moments of aggregate discounted claims, where the dependence between the inter-claim time and the subsequent claim size is considered. Using the general expression for the m-th order moment proposed by Léveillé and Garrido (Scand. Actuar. J. 2001 , 2, 98–110), which takes the form of the Volterra integral equation (VIE), we used the method of successive approximation to derive the Neumann series of the recursive moments. We then compute the first two moments of aggregate discounted claims, i.e., its mean and variance, based on the Neumann series expression, where the dependence structure is captured by a Farlie–Gumbel–Morgenstern (FGM) copula, a Gaussian copula and a Gumbel copula with exponential marginal distributions. Insurance premium calculations with their figures are also illustrated.

Suggested Citation

  • Siti Norafidah Mohd Ramli & Jiwook Jang, 2014. "Neumann Series on the Recursive Moments of Copula-Dependent Aggregate Discounted Claims," Risks, MDPI, vol. 2(2), pages 1-16, May.
    • Handle: RePEc:gam:jrisks:v:2:y:2014:i:2:p:195-210:d:36506
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    References listed on IDEAS

    as
    1. 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.
    2. Bargès, Mathieu & Cossette, Hélène & Loisel, Stéphane & Marceau, Étienne, 2011. "On the Moments of Aggregate Discounted Claims with Dependence Introduced by a FGM Copula," ASTIN Bulletin, Cambridge University Press, vol. 41(1), pages 215-238, May.
    3. Ji‐Wook Jang, 2004. "Martingale Approach for Moments of Discounted Aggregate Claims," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 71(2), pages 201-211, June.
    4. 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.
    5. Woo, Jae-Kyung, 2010. "Some Remarks on Delayed Renewal Risk Models," ASTIN Bulletin, Cambridge University Press, vol. 40(1), pages 199-219, May.
    6. Albrecher, Hansjorg & Boxma, Onno J., 2004. "A ruin model with dependence between claim sizes and claim intervals," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 245-254, October.
    7. Woo, Jae-Kyung & Cheung, Eric C.K., 2013. "A note on discounted compound renewal sums under dependency," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 170-179.
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    Cited by:

    1. Shuanming Li & Yi Lu, 2018. "On the Moments and the Distribution of Aggregate Discounted Claims in a Markovian Environment," Risks, MDPI, vol. 6(2), pages 1-16, May.
    2. Ya Fang Wang & José Garrido & Ghislain Léveillé, 2018. "The Distribution of Discounted Compound PH–Renewal Processes," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 69-96, March.
    3. 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.
    4. Sharifah Farah Syed Yusoff Alhabshi & Zamira Hasanah Zamzuri & Siti Norafidah Mohd Ramli, 2021. "Monte Carlo Simulation of the Moments of a Copula-Dependent Risk Process with Weibull Interwaiting Time," Risks, MDPI, vol. 9(6), pages 1-21, June.

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