IDEAS home Printed from https://ideas.repec.org/a/gam/jrisks/v5y2017i1p3-d87055.html
   My bibliography  Save this article

The Effects of Largest Claim and Excess of Loss Reinsurance on a Company’s Ruin Time and Valuation

Author

Listed:
  • Yuguang Fan

    (ARC Centre of Excellence for Mathematical and Statistical Frontiers, School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia
    Research School of Finance, Actuarial Studies and Statistics, Australian National University, Canberra, ACT 0200, Australia)

  • Philip S. Griffin

    (Department of Mathematics, Syracuse University, Syracuse, NY 13244-1150, USA)

  • Ross Maller

    (Research School of Finance, Actuarial Studies and Statistics, Australian National University, Canberra, ACT 0200, Australia)

  • Alexander Szimayer

    (School of Economics and Social Science, Universität Hamburg, Von-Melle-Park 5, 20146 Hamburg, Germany)

  • Tiandong Wang

    (School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, USA)

Abstract

We compare two types of reinsurance: excess of loss (EOL) and largest claim reinsurance (LCR), each of which transfers the payment of part, or all, of one or more large claims from the primary insurance company (the cedant) to a reinsurer. The primary insurer’s point of view is documented in terms of assessment of risk and payment of reinsurance premium. A utility indifference rationale based on the expected future dividend stream is used to value the company with and without reinsurance. Assuming the classical compound Poisson risk model with choices of claim size distributions (classified as heavy, medium and light-tailed cases), simulations are used to illustrate the impact of the EOL and LCR treaties on the company’s ruin probability, ruin time and value as determined by the dividend discounting model. We find that LCR is at least as effective as EOL in averting ruin in comparable finite time horizon settings. In instances where the ruin probability for LCR is smaller than for EOL, the dividend discount model shows that the cedant is able to pay a larger portion of the dividend for LCR reinsurance than for EOL while still maintaining company value. Both methods reduce risk considerably as compared with no reinsurance, in a variety of situations, as measured by the standard deviation of the company value. A further interesting finding is that heaviness of tails alone is not necessarily the decisive factor in the possible ruin of a company; small and moderate sized claims can also play a significant role in this.

Suggested Citation

  • Yuguang Fan & Philip S. Griffin & Ross Maller & Alexander Szimayer & Tiandong Wang, 2017. "The Effects of Largest Claim and Excess of Loss Reinsurance on a Company’s Ruin Time and Valuation," Risks, MDPI, vol. 5(1), pages 1-27, January.
    • Handle: RePEc:gam:jrisks:v:5:y:2017:i:1:p:3-:d:87055
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-9091/5/1/3/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-9091/5/1/3/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Neil Doherty & Kent Smetters, 2005. "Moral Hazard in Reinsurance Markets," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 72(3), pages 375-391, September.
    2. Benktander, G., 1978. "Largest Claims Reinsurance (LCR). A Quick Method to Calculate LCR-Risk Rates from Excess of Loss Risk Rates," ASTIN Bulletin, Cambridge University Press, vol. 10(1), pages 54-58, May.
    3. Stéphane Loisel & Hans-U. Gerber, 2012. "Why ruin theory should be of interest for insurance practitioners and risk managers nowadays," Post-Print hal-00746231, HAL.
    4. Vytaras Brazauskas & Andreas Kleefeld, 2016. "Modeling Severity and Measuring Tail Risk of Norwegian Fire Claims," North American Actuarial Journal, Taylor & Francis Journals, vol. 20(1), pages 1-16, January.
    5. Kremer, Erhard, 1983. "Distribution-free upper bounds on the premiums of the LCR and ECOMOR treaties," Insurance: Mathematics and Economics, Elsevier, vol. 2(3), pages 209-213, July.
    6. Kremer, Erhard, 1990. "The Asymptotic Efficiency of Largest Claims Reinsurance Treaties," ASTIN Bulletin, Cambridge University Press, vol. 20(1), pages 11-22, April.
    7. Griffin, Philip S. & Maller, Ross A. & Schaik, Kees van, 2012. "Asymptotic distributions of the overshoot and undershoots for the Lévy insurance risk process in the Cramér and convolution equivalent cases," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 382-392.
    8. Wikstad, Nils, 1971. "Exemplification of Ruin Probabilities," ASTIN Bulletin, Cambridge University Press, vol. 6(2), pages 147-152, December.
    9. Dickson, David C.M. & Waters, Howard R., 2004. "Some Optimal Dividends Problems," ASTIN Bulletin, Cambridge University Press, vol. 34(1), pages 49-74, May.
    10. Griffin, Philip S. & Maller, Ross A. & Roberts, Dale, 2013. "Finite time ruin probabilities for tempered stable insurance risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 478-489.
    11. Kremer, Erhard, 1998. "Largest Claims Reinsurance Premiums under Possible Claims Dependence," ASTIN Bulletin, Cambridge University Press, vol. 28(2), pages 257-267, November.
    12. Berglund, Raoul M., 1998. "A Note on the Net Premium for a Generalized Largest Claims Reinsurance Cover," ASTIN Bulletin, Cambridge University Press, vol. 28(1), pages 153-162, May.
    13. Asmussen, S. & Binswanger, K., 1997. "Simulation of Ruin Probabilities for Subexponential Claims," ASTIN Bulletin, Cambridge University Press, vol. 27(2), pages 297-318, November.
    14. Kremer, Erhard, 1982. "Rating of Largest Claims and Ecomor Reinsurance Treaties for Large Portfolios," ASTIN Bulletin, Cambridge University Press, vol. 13(1), pages 47-56, June.
    15. 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. Ipsen, Yuguang & Maller, Ross & Resnick, Sidney, 2019. "Ratios of ordered points of point processes with regularly varying intensity measures," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 205-222.

    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. & Peralta, Oscar & Woo, Jae-Kyung, 2022. "Multivariate matrix-exponential affine mixtures and their applications in risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 364-389.
    2. Castaño-Martínez, A. & Pigueiras, G. & Sordo, M.A., 2019. "On a family of risk measures based on largest claims," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 92-97.
    3. Eric C. K. Cheung & Oscar Peralta & Jae-Kyung Woo, 2021. "Multivariate matrix-exponential affine mixtures and their applications in risk theory," Papers 2201.11122, arXiv.org.
    4. Hess, Christian, 2009. "Computing the mean and the variance of the cedent's share for largest claims reinsurance covers," Insurance: Mathematics and Economics, Elsevier, vol. 44(3), pages 497-504, June.
    5. repec:dau:papers:123456789/4715 is not listed on IDEAS
    6. Robert Hartwig & Greg Niehaus & Joseph Qiu, 2020. "Insurance for economic losses caused by pandemics," The Geneva Risk and Insurance Review, Palgrave Macmillan;International Association for the Study of Insurance Economics (The Geneva Association), vol. 45(2), pages 134-170, September.
    7. Albrecher Hansjörg & Kantor Josef, 2002. "Simulation of ruin probabilities for risk processes of Markovian type," Monte Carlo Methods and Applications, De Gruyter, vol. 8(2), pages 111-128, December.
    8. Hangsuck Lee & Minha Lee & Jimin Hong, 2024. "Optimal insurance for repetitive natural disasters under moral hazard," Journal of Economics, Springer, vol. 143(3), pages 247-277, December.
    9. Grandell, Jan, 2000. "Simple approximations of ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 26(2-3), pages 157-173, May.
    10. Dassios, Angelos & Jang, Jiwook & Zhao, Hongbiao, 2015. "A risk model with renewal shot-noise Cox process," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 55-65.
    11. De Vylder, F. Etienne & Goovaerts, Marc J., 1999. "Explicit finite-time and infinite-time ruin probabilities in the continuous case," Insurance: Mathematics and Economics, Elsevier, vol. 24(3), pages 155-172, May.
    12. Dassios, Angelos & Wu, Shanle, 2008. "Parisian ruin with exponential claims," LSE Research Online Documents on Economics 32033, London School of Economics and Political Science, LSE Library.
    13. Xu, Ran & Woo, Jae-Kyung, 2020. "Optimal dividend and capital injection strategy with a penalty payment at ruin: Restricted dividend payments," Insurance: Mathematics and Economics, Elsevier, vol. 92(C), pages 1-16.
    14. Kaluszka, Marek & Okolewski, Andrzej, 2011. "Stability of L-statistics from weakly dependent observations," Statistics & Probability Letters, Elsevier, vol. 81(5), pages 618-625, May.
    15. Vierkötter, Matthias & Schmidli, Hanspeter, 2017. "On optimal dividends with exponential and linear penalty payments," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 265-270.
    16. Peng, Liang, 2014. "Joint tail of ECOMOR and LCR reinsurance treaties," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 116-120.
    17. Vaios Dermitzakis & Konstadinos Politis, 2011. "Asymptotics for the Moments of the Time to Ruin for the Compound Poisson Model Perturbed by Diffusion," Methodology and Computing in Applied Probability, Springer, vol. 13(4), pages 749-761, December.
    18. Luis Rincón & David J. Santana, 2022. "Ruin Probability for Finite Erlang Mixture Claims Via Recurrence Sequences," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 2213-2236, September.
    19. Benjamin Avanzi & Jos'e-Luis P'erez & Bernard Wong & Kazutoshi Yamazaki, 2016. "On optimal joint reflective and refractive dividend strategies in spectrally positive L\'evy models," Papers 1607.01902, arXiv.org, revised Nov 2016.
    20. Reynkens, Tom & Verbelen, Roel & Beirlant, Jan & Antonio, Katrien, 2017. "Modelling censored losses using splicing: A global fit strategy with mixed Erlang and extreme value distributions," Insurance: Mathematics and Economics, Elsevier, vol. 77(C), pages 65-77.
    21. Eisenberg, Julia & Krühner, Paul, 2018. "The impact of negative interest rates on optimal capital injections," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 1-10.

    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:gam:jrisks:v:5:y:2017:i:1:p:3-:d:87055. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.