IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v52y2013i2p170-179.html
   My bibliography  Save this article

A note on discounted compound renewal sums under dependency

Author

Listed:
  • Woo, Jae-Kyung
  • Cheung, Eric C.K.

Abstract

The paper considers a renewal risk process in which a given inter-arrival time possibly has an impact on the size of the resulting claim. Under a fairly general dependency structure which contains various well-known examples in the literature as special cases, recursive formulas for the moments of the discounted aggregate claims are derived using the techniques in Léveillé and Garrido (2001b). Simplifications arise in the case of a dependent renewal risk process under ‘Erlang weights’. Numerical examples are given towards the end to illustrate the impact of dependency on the discounted aggregate claims.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:insuma:v:52:y:2013:i:2:p:170-179
    DOI: 10.1016/j.insmatheco.2012.11.005
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668712001436
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.insmatheco.2012.11.005?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Rong Wu & Yuhua Lu & Ying Fang, 2007. "On the Gerber-Shiu Discounted Penalty Function for the Ordinary Renewal Risk Model with Constant Interest," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(2), pages 119-134.
    2. Jiandong Ren, 2008. "On the Laplace Transform of the Aggregate Discounted Claims with Markovian Arrivals," North American Actuarial Journal, Taylor & Francis Journals, vol. 12(2), pages 198-206.
    3. Cai, Jun & Dickson, David C. M., 2003. "Upper bounds for ultimate ruin probabilities in the Sparre Andersen model with interest," Insurance: Mathematics and Economics, Elsevier, vol. 32(1), pages 61-71, February.
    4. Sundt, Bjorn & Teugels, Jozef L., 1995. "Ruin estimates under interest force," Insurance: Mathematics and Economics, Elsevier, vol. 16(1), pages 7-22, April.
    5. 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.
    6. Daniel Dufresne, 2007. "Fitting combinations of exponentials to probability distributions," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 23(1), pages 23-48, January.
    7. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    8. Ambagaspitiya, Rohana S., 2009. "Ultimate ruin probability in the Sparre Andersen model with dependent claim sizes and claim occurrence times," Insurance: Mathematics and Economics, Elsevier, vol. 44(3), pages 464-472, June.
    9. I. Bairamov & S. Kotz & M. Bekci, 2001. "New generalized Farlie-Gumbel-Morgenstern distributions and concomitants of order statistics," Journal of Applied Statistics, Taylor & Francis Journals, vol. 28(5), pages 521-536.
    10. Rodríguez-Lallena, José Antonio & Úbeda-Flores, Manuel, 2004. "A new class of bivariate copulas," Statistics & Probability Letters, Elsevier, vol. 66(3), pages 315-325, February.
    11. Dickson, David C. M. & Hipp, Christian, 2001. "On the time to ruin for Erlang(2) risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 29(3), pages 333-344, December.
    12. Cai, Jun & Dickson, David C. M., 2002. "On the expected discounted penalty function at ruin of a surplus process with interest," Insurance: Mathematics and Economics, Elsevier, vol. 30(3), pages 389-404, June.
    13. Delbaen, F. & Haezendonck, J., 1987. "Classical risk theory in an economic environment," Insurance: Mathematics and Economics, Elsevier, vol. 6(2), pages 85-116, April.
    14. 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.
    15. 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.
    16. Woo, Jae-Kyung, 2010. "Some Remarks on Delayed Renewal Risk Models," ASTIN Bulletin, Cambridge University Press, vol. 40(1), pages 199-219, May.
    17. 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.
    18. 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.
    19. Willmot, Gordon E. & Woo, Jae-Kyung, 2012. "On the analysis of a general class of dependent risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 134-141.
    20. Sancetta, Alessio & Satchell, Stephen, 2004. "The Bernstein Copula And Its Applications To Modeling And Approximations Of Multivariate Distributions," Econometric Theory, Cambridge University Press, vol. 20(3), pages 535-562, June.
    21. Cossette, Hélène & Marceau, Etienne & Marri, Fouad, 2008. "On the compound Poisson risk model with dependence based on a generalized Farlie-Gumbel-Morgenstern copula," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 444-455, December.
    22. 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.
    23. Li, Shuanming & Garrido, Jose, 2004. "On ruin for the Erlang(n) risk process," Insurance: Mathematics and Economics, Elsevier, vol. 34(3), pages 391-408, June.
    24. Kalashnikov, Vladimir & Konstantinides, Dimitrios, 2000. "Ruin under interest force and subexponential claims: a simple treatment," Insurance: Mathematics and Economics, Elsevier, vol. 27(1), pages 145-149, August.
    25. Dickson, D. C. M., 2001. "Lundberg Approximations for Compound Distributions with Insurance Applications. By G. E. Willmot and X. S. Lin. (Springer, 2000)," British Actuarial Journal, Cambridge University Press, vol. 7(4), pages 690-691, October.
    26. Cheung, Eric C.K., 2011. "A generalized penalty function in Sparre Andersen risk models with surplus-dependent premium," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 384-397, May.
    27. Diers, Dorothea & Eling, Martin & Marek, Sebastian D., 2012. "Dependence modeling in non-life insurance using the Bernstein copula," Insurance: Mathematics and Economics, Elsevier, vol. 50(3), pages 430-436.
    28. Cheung, Eric C.K. & Landriault, David & Willmot, Gordon E. & Woo, Jae-Kyung, 2010. "Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 117-126, February.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Landriault, David & Willmot, Gordon E. & Xu, Di, 2014. "On the analysis of time dependent claims in a class of birth process claim count models," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 168-173.
    2. Landy Rabehasaina & Jae-Kyung Woo, 2018. "On a multivariate renewal-reward process involving time delays and discounting: applications to IBNR processes and infinite server queues," Queueing Systems: Theory and Applications, Springer, vol. 90(3), pages 307-350, December.
    3. Cheung, Eric C.K. & Ni, Weihong & Oh, Rosy & Woo, Jae-Kyung, 2021. "Bayesian credibility under a bivariate prior on the frequency and the severity of claims," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 274-295.
    4. Blier-Wong, Christopher & Cossette, Hélène & Marceau, Etienne, 2023. "Risk aggregation with FGM copulas," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 102-120.
    5. Woo, Jae-Kyung, 2016. "On multivariate discounted compound renewal sums with time-dependent claims in the presence of reporting/payment delays," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 354-363.
    6. 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.
    7. Ghislain Léveillé & Ilie-Radu Mitric & Victor Côté, 2018. "Effects of the Age Process on Aggregate Discounted Claims," Risks, MDPI, vol. 6(4), pages 1-17, September.
    8. Castañer, A. & Claramunt, M.M. & Lefèvre, C. & Loisel, S., 2015. "Discrete Schur-constant models," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 343-362.
    9. 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.
    10. 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.
    11. Castañer, A. & Claramunt, M.M. & Lefèvre, C. & Loisel, S., 2015. "Discrete Schur-constant models," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 343-362.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Cheung, Eric C.K., 2013. "Moments of discounted aggregate claim costs until ruin in a Sparre Andersen risk model with general interclaim times," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 343-354.
    2. Zhimin Zhang & Hailiang Yang & Hu Yang, 2012. "On a Sparre Andersen Risk Model with Time-Dependent Claim Sizes and Jump-Diffusion Perturbation," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 973-995, December.
    3. He, Yue & Kawai, Reiichiro & Shimizu, Yasutaka & Yamazaki, Kazutoshi, 2023. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 109(C), pages 1-28.
    4. Willmot, Gordon E. & Woo, Jae-Kyung, 2012. "On the analysis of a general class of dependent risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 134-141.
    5. Cheung, Eric C.K., 2011. "A generalized penalty function in Sparre Andersen risk models with surplus-dependent premium," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 384-397, May.
    6. 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.
    7. Cheung, Eric C.K. & Liu, Haibo & Willmot, Gordon E., 2018. "Joint moments of the total discounted gains and losses in the renewal risk model with two-sided jumps," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 358-377.
    8. Eric C.K. Cheung & Haibo Liu & Jae-Kyung Woo, 2015. "On the Joint Analysis of the Total Discounted Payments to Policyholders and Shareholders: Dividend Barrier Strategy," Risks, MDPI, vol. 3(4), pages 1-24, November.
    9. Lee, Wing Yan & Willmot, Gordon E., 2014. "On the moments of the time to ruin in dependent Sparre Andersen models with emphasis on Coxian interclaim times," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 1-10.
    10. Jostein Paulsen, 2008. "Ruin models with investment income," Papers 0806.4125, arXiv.org, revised Dec 2008.
    11. Franck Adékambi & Kokou Essiomle, 2021. "Asymptotic Tail Probability of the Discounted Aggregate Claims under Homogeneous, Non-Homogeneous and Mixed Poisson Risk Model," Risks, MDPI, vol. 9(7), pages 1-22, June.
    12. Wong, Jeff T.Y. & Cheung, Eric C.K., 2015. "On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 280-290.
    13. 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.
    14. Sheldon Lin, X. & E. Willmot, Gordon & Drekic, Steve, 2003. "The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function," Insurance: Mathematics and Economics, Elsevier, vol. 33(3), pages 551-566, December.
    15. Landriault, David, 2008. "Constant dividend barrier in a risk model with interclaim-dependent claim sizes," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 31-38, February.
    16. Hélène Cossette & Etienne Marceau & Fouad Marri, 2011. "Constant Dividend Barrier in a Risk Model with a Generalized Farlie-Gumbel-Morgenstern Copula," Methodology and Computing in Applied Probability, Springer, vol. 13(3), pages 487-510, September.
    17. Yue He & Reiichiro Kawai & Yasutaka Shimizu & Kazutoshi Yamazaki, 2022. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Papers 2203.10680, arXiv.org, revised Dec 2022.
    18. Huynh, Mirabelle & Landriault, David & Shi, Tianxiang & Willmot, Gordon E., 2015. "On a risk model with claim investigation," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 37-45.
    19. Heilpern, Stanislaw, 2014. "Ruin measures for a compound Poisson risk model with dependence based on the Spearman copula and the exponential claim sizes," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 251-257.
    20. Cheung, Eric C.K. & Feng, Runhuan, 2013. "A unified analysis of claim costs up to ruin in a Markovian arrival risk model," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 98-109.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:insuma:v:52:y:2013:i:2:p:170-179. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/inca/505554 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.