IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v51y2012i2p393-401.html
   My bibliography  Save this article

Analysis of the discounted sum of ascending ladder heights

Author

Listed:
  • Cossette, Hélène
  • Landriault, David
  • Marceau, Etienne
  • Moutanabbir, Khouzeima

Abstract

Within the Sparre Andersen risk model, the ruin probability corresponds to the survival function of the maximal aggregate loss. It is well known that the maximum aggregate loss follows a compound geometric distribution, in which the summands consist of the ascending ladder heights. In the present paper, we propose to investigate the distribution of the discounted sum of ascending ladder heights over a finite or an infinite-time intervals. In particular, the moments of the discounted sum of ascending ladder heights over a finite and an infinite-time intervals are derived in both the classical compound Poisson risk model and the Sparre Andersen risk model with exponential claims. The application of a particular Gerber–Shiu functional is central to the derivation of these results, as is the mixed Erlang distributional assumption. Finally, we define VaR and TVaR risk measures in terms of the discounted sum of ascending ladder heights. We use a moment-matching method to approximate the distribution of the discounted sum of ascending ladder heights allowing the computation of the VaR and TVaR risk measures. We conclude this paper with a numerical example illustrating different topics discussed in the paper.

Suggested Citation

  • Cossette, Hélène & Landriault, David & Marceau, Etienne & Moutanabbir, Khouzeima, 2012. "Analysis of the discounted sum of ascending ladder heights," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 393-401.
    • Handle: RePEc:eee:insuma:v:51:y:2012:i:2:p:393-401
    DOI: 10.1016/j.insmatheco.2012.06.004
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668712000716
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.insmatheco.2012.06.004?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. Gordon E. Willmot & X. Sheldon Lin, 2011. "Risk modelling with the mixed Erlang distribution," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 27(1), pages 2-16, January.
    2. Borovkov, Konstantin A. & Dickson, David C.M., 2008. "On the ruin time distribution for a Sparre Andersen process with exponential claim sizes," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 1104-1108, June.
    3. Acerbi, Carlo, 2002. "Spectral measures of risk: A coherent representation of subjective risk aversion," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1505-1518, July.
    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. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
    6. 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.
    7. Gordon Willmot & Jae-Kyung Woo, 2007. "On the Class of Erlang Mixtures with Risk Theoretic Applications," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(2), pages 99-115.
    8. David Landriault & Gordon Willmot, 2009. "On the Joint Distributions of the Time to Ruin, the Surplus Prior to Ruin, and the Deficit at Ruin in the Classical Risk Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 13(2), pages 252-270.
    9. Lin, X. Sheldon & Willmot, Gordon E., 2000. "The moments of the time of ruin, the surplus before ruin, and the deficit at ruin," Insurance: Mathematics and Economics, Elsevier, vol. 27(1), pages 19-44, August.
    10. Landriault, David & Shi, Tianxiang & Willmot, Gordon E., 2011. "Joint densities involving the time to ruin in the Sparre Andersen risk model under exponential assumptions," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 371-379.
    11. Jiandong Ren, 2007. "The Discounted Joint Distribution of the Surplus Prior to Ruin and the Deficit at Ruin in a Sparre Andersen Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(3), pages 128-136.
    12. Lin, X. Sheldon & Willmot, Gordon E., 1999. "Analysis of a defective renewal equation arising in ruin theory," Insurance: Mathematics and Economics, Elsevier, vol. 25(1), pages 63-84, September.
    13. Trufin, J. & Albrecher, H. & Denuit, M., 2010. "Properties of risk measures derived from ruin theory," LIDAM Discussion Papers ISBA 2010019, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    14. Dickson,David C. M., 2010. "Insurance Risk and Ruin," Cambridge Books, Cambridge University Press, number 9780521176750.
    15. Tsai, Cary Chi-Liang & Willmot, Gordon E., 2002. "On the moments of the surplus process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 31(3), pages 327-350, December.
    16. Mathieu Bargès & Hélène Cossette & Etienne Marceau, 2009. "TVaR-based capital allocation with copulas," Working Papers hal-00431265, HAL.
    17. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    18. Bargès, Mathieu & Cossette, Hélène & Marceau, Étienne, 2009. "TVaR-based capital allocation with copulas," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 348-361, December.
    19. Dickson, David C.M. & Willmot, Gordon E., 2005. "The Density of the Time to Ruin in the Classical Poisson Risk Model," ASTIN Bulletin, Cambridge University Press, vol. 35(1), pages 45-60, May.
    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. Cossette, Hélène & Côté, Marie-Pier & Marceau, Etienne & Moutanabbir, Khouzeima, 2013. "Multivariate distribution defined with Farlie–Gumbel–Morgenstern copula and mixed Erlang marginals: Aggregation and capital allocation," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 560-572.

    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. Cossette, Hélène & Mailhot, Mélina & Marceau, Étienne, 2012. "TVaR-based capital allocation for multivariate compound distributions with positive continuous claim amounts," Insurance: Mathematics and Economics, Elsevier, vol. 50(2), pages 247-256.
    2. Landriault, David & Lemieux, Christiane & Willmot, Gordon E., 2012. "An adaptive premium policy with a Bayesian motivation in the classical risk model," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 370-378.
    3. Landriault, David & Shi, Tianxiang & Willmot, Gordon E., 2011. "Joint densities involving the time to ruin in the Sparre Andersen risk model under exponential assumptions," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 371-379.
    4. Landriault, David & Li, Bin & Shi, Tianxiang & Xu, Di, 2019. "On the distribution of classic and some exotic ruin times," Insurance: Mathematics and Economics, Elsevier, vol. 89(C), pages 38-45.
    5. Feng, Runhuan & Volkmer, Hans W., 2012. "Modeling credit value adjustment with downgrade-triggered termination clause using a ruin theoretic approach," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 409-421.
    6. 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.
    7. 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.
    8. Li, Shuanming & Lu, Yi, 2017. "Distributional study of finite-time ruin related problems for the classical risk model," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 319-330.
    9. Blier-Wong, Christopher & Cossette, Hélène & Marceau, Etienne, 2023. "Risk aggregation with FGM copulas," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 102-120.
    10. Tang, Qihe & Wei, Li, 2010. "Asymptotic aspects of the Gerber-Shiu function in the renewal risk model using Wiener-Hopf factorization and convolution equivalence," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 19-31, February.
    11. Philipp Lukas Strietzel & Anita Behme, 2022. "Moments of the Ruin Time in a Lévy Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 3075-3099, December.
    12. Li, Jingchao & Dickson, David C.M. & Li, Shuanming, 2015. "Some ruin problems for the MAP risk model," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 1-8.
    13. 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.
    14. 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.
    15. Frostig, Esther & Pitts, Susan M. & Politis, Konstadinos, 2012. "The time to ruin and the number of claims until ruin for phase-type claims," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 19-25.
    16. Dickson, David C.M. & Li, Shuanming, 2010. "Finite time ruin problems for the Erlang(2) risk model," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 12-18, February.
    17. Cossette, Hélène & Marceau, Etienne & Mtalai, Itre, 2019. "Collective risk models with dependence," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 153-168.
    18. Dickson, David C.M., 2012. "The joint distribution of the time to ruin and the number of claims until ruin in the classical risk model," Insurance: Mathematics and Economics, Elsevier, vol. 50(3), pages 334-337.
    19. Cheung, Eric C.K. & Zhu, Wei, 2023. "Cumulative Parisian ruin in finite and infinite time horizons for a renewal risk process with exponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 84-101.
    20. Michael V. Boutsikas & Konstadinos Politis, 2017. "Exit Times, Overshoot and Undershoot for a Surplus Process in the Presence of an Upper Barrier," Methodology and Computing in Applied Probability, Springer, vol. 19(1), pages 75-95, March.

    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:51:y:2012:i:2:p:393-401. 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.