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On log-normal convolutions: An analytical–numerical method with applications to economic capital determination

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  • Furman, Edward
  • Hackmann, Daniel
  • Kuznetsov, Alexey

Abstract

We put forward an efficient algorithm for approximating the sums of independent and log-normally distributed random variables. Namely, by combining tools from probability theory and numerical analysis, we are able to compute the cumulative distribution functions of the just-mentioned sums to a high precision and in a relatively short computing time. We illustrate the effectiveness of the new method in the contexts of the individual and collective risk models, aggregate economic capital determination, and economic capital allocation.

Suggested Citation

  • Furman, Edward & Hackmann, Daniel & Kuznetsov, Alexey, 2020. "On log-normal convolutions: An analytical–numerical method with applications to economic capital determination," Insurance: Mathematics and Economics, Elsevier, vol. 90(C), pages 120-134.
    • Handle: RePEc:eee:insuma:v:90:y:2020:i:c:p:120-134
    DOI: 10.1016/j.insmatheco.2019.10.003
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    References listed on IDEAS

    as
    1. Gary Venter, 2004. "Capital Allocation Survey with Commentary," North American Actuarial Journal, Taylor & Francis Journals, vol. 8(2), pages 96-107.
    2. Denuit, Michel & Dhaene, Jan & Ribas, Carmen, 2001. "Does positive dependence between individual risks increase stop-loss premiums?," Insurance: Mathematics and Economics, Elsevier, vol. 28(3), pages 305-308, June.
    3. Rob Kaas & Marc Goovaerts & Jan Dhaene & Michel Denuit, 2008. "Modern Actuarial Risk Theory," Springer Books, Springer, edition 2, number 978-3-540-70998-5, June.
    4. Jan Dhaene & Andreas Tsanakas & Emiliano A. Valdez & Steven Vanduffel, 2012. "Optimal Capital Allocation Principles," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 79(1), pages 1-28, March.
    5. Søren Asmussen & Jens Ledet Jensen & Leonardo Rojas-Nandayapa, 2016. "Exponential Family Techniques for the Lognormal Left Tail," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(3), pages 774-787, September.
    6. Moshe Arye Milevsky & Steven E. Posner, 1999. "Asian Options, The Sum Of Lognormals, And The Reciprocal Gamma Distribution," World Scientific Book Chapters, in: Marco Avellaneda (ed.), Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar, chapter 7, pages 203-218, World Scientific Publishing Co. Pte. Ltd..
    7. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    8. Asmussen, Søren, 2018. "Conditional Monte Carlo for sums, with applications to insurance and finance," Annals of Actuarial Science, Cambridge University Press, vol. 12(2), pages 455-478, September.
    9. Furman, Edward & Zitikis, Ricardas, 2008. "Weighted premium calculation principles," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 459-465, February.
    10. Wang, Shaun, 1996. "Premium Calculation by Transforming the Layer Premium Density," ASTIN Bulletin, Cambridge University Press, vol. 26(1), pages 71-92, May.
    11. Archil Gulisashvili & Peter Tankov, 2013. "Tail behavior of sums and differences of log-normal random variables," Papers 1309.3057, arXiv.org, revised Jan 2016.
    12. Furman, Edward & Landsman, Zinoviy, 2006. "Tail Variance Premium with Applications for Elliptical Portfolio of Risks," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 433-462, November.
    13. Dhaene, J. & Henrard, L. & Landsman, Z. & Vandendorpe, A. & Vanduffel, S., 2008. "Some results on the CTE-based capital allocation rule," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 855-863, April.
    14. Søren Asmussen & Jens Ledet Jensen & Leonardo Rojas-Nandayapa, 2016. "On the Laplace Transform of the Lognormal Distribution," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 441-458, June.
    15. Asmussen, Søren & Rojas-Nandayapa, Leonardo, 2008. "Asymptotics of sums of lognormal random variables with Gaussian copula," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2709-2714, November.
    16. Furman, Edward & Landsman, Zinoviy, 2005. "Risk capital decomposition for a multivariate dependent gamma portfolio," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 635-649, December.
    17. Furman, Edward & Zitikis, Ricardas, 2008. "Weighted risk capital allocations," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 263-269, October.
    18. Jiang, Chun-Fu & Peng, Hong-Yi & Yang, Yu-Kuan, 2016. "Tail variance of portfolio under generalized Laplace distribution," Applied Mathematics and Computation, Elsevier, vol. 282(C), pages 187-203.
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    Cited by:

    1. Boyle, Phelim & Jiang, Ruihong, 2023. "A note on portfolios of averages of lognormal variables," Insurance: Mathematics and Economics, Elsevier, vol. 112(C), pages 97-109.
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    3. Nawaf Mohammed & Edward Furman & Jianxi Su, 2021. "Can a regulatory risk measure induce profit-maximizing risk capital allocations? The case of Conditional Tail Expectation," Papers 2102.05003, arXiv.org, revised Aug 2021.
    4. Gao, Guangyuan, 2024. "Fitting Tweedie's compound Poisson model to pure premium with the EM algorithm," Insurance: Mathematics and Economics, Elsevier, vol. 114(C), pages 29-42.

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    More about this item

    Keywords

    Log-normal distribution; Convolution; Generalized gamma convolution; Padé approximation; Individual risk model; Collective risk model; Economic capital;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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