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Multiplicative background risk models: Setting a course for the idiosyncratic risk factors distributed phase-type

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  • Furman, Edward
  • Kye, Yisub
  • Su, Jianxi

Abstract

Multiplicative background risk models in which the idiosyncratic risk factors are assumed to be distributed exponentially, and the systemic risk factor has an arbitrary distribution on the non-negative half of the real line have seen a great variety of applications in actuarial science. Admittedly, these structures, which are well-known to mathematical statisticians under the name of exponential mixtures, enjoy remarkable level of technical tractability and so are a convenient tool for modelling risk components in a portfolio of an insurer. That said, the assumption of exponentiality is merely a mathematical nicety and does not have to reflect reality, yet the works that loosen this assumption are rare. The goals of this paper are two-fold. Firstly, we pursue a holistic approach and discuss in detail the multiplicative background risk models with arbitrarily distributed idiosyncratic and systemic risk factors. In this respect, we systematize the existing results and report some new ones. Secondly, and more importantly, we focus on the special case when the distribution of the idiosyncratic risk factors is phase-type. The novel theory, which allows to introduce significant heterogeneity in the idiosyncratic risk factors, is illustrated by numerous numerical examples borrowed from the context of the determination and allocation of economic capital. The examples suggest that a little departure from exponentiality can have substantial impact on the outcome of risk analysis.

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  • Furman, Edward & Kye, Yisub & Su, Jianxi, 2021. "Multiplicative background risk models: Setting a course for the idiosyncratic risk factors distributed phase-type," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 153-167.
    • Handle: RePEc:eee:insuma:v:96:y:2021:i:c:p:153-167
    DOI: 10.1016/j.insmatheco.2020.11.007
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    1. Albrecher, Hansjörg & Constantinescu, Corina & Loisel, Stephane, 2011. "Explicit ruin formulas for models with dependence among risks," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 265-270, March.
    2. Stéphane Loisel, 2011. "On ruin models with dependent risks," Post-Print hal-00671926, HAL.
    3. Furman, Edward & Zitikis, Ricardas, 2008. "Weighted premium calculation principles," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 459-465, February.
    4. Wang, Shaun, 1996. "Premium Calculation by Transforming the Layer Premium Density," ASTIN Bulletin, Cambridge University Press, vol. 26(1), pages 71-92, May.
    5. Vadim Semenikhine & Edward Furman & Jianxi Su, 2018. "On a Multiplicative Multivariate Gamma Distribution with Applications in Insurance," Risks, MDPI, vol. 6(3), pages 1-20, August.
    6. Stephen A. Ross, 2013. "The Arbitrage Theory of Capital Asset Pricing," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 1, pages 11-30, World Scientific Publishing Co. Pte. Ltd..
    7. Gordy, Michael B., 2003. "A risk-factor model foundation for ratings-based bank capital rules," Journal of Financial Intermediation, Elsevier, vol. 12(3), pages 199-232, July.
    8. Cossette, Hélène & Marceau, Etienne & Mtalai, Itre & Veilleux, Déry, 2018. "Dependent risk models with Archimedean copulas: A computational strategy based on common mixtures and applications," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 53-71.
    9. Su, Jianxi & Furman, Edward, 2017. "Multiple risk factor dependence structures: Distributional properties," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 56-68.
    10. Raluca Vernic, 2011. "Tail Conditional Expectation for the Multivariate Pareto Distribution of the Second Kind: Another Approach," Methodology and Computing in Applied Probability, Springer, vol. 13(1), pages 121-137, March.
    11. Sanjiv R. Das & Darrell Duffie & Nikunj Kapadia & Leandro Saita, 2007. "Common Failings: How Corporate Defaults Are Correlated," Journal of Finance, American Finance Association, vol. 62(1), pages 93-117, February.
    12. Alexandru V. Asimit & Raluca Vernic & Ricardas Zitikis, 2016. "Background Risk Models and Stepwise Portfolio Construction," Methodology and Computing in Applied Probability, Springer, vol. 18(3), pages 805-827, September.
    13. Côté, Marie-Pier & Genest, Christian, 2019. "Dependence in a background risk model," Journal of Multivariate Analysis, Elsevier, vol. 172(C), pages 28-46.
    14. 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.
    15. Furman, Edward & Zitikis, RiÄ ardas, 2010. "General Stein-Type Covariance Decompositions with Applications to Insurance and Finance," ASTIN Bulletin, Cambridge University Press, vol. 40(1), pages 369-375, May.
    16. Edward Furman & Ričardas Zitikis, 2009. "Weighted Pricing Functionals With Applications to Insurance," North American Actuarial Journal, Taylor & Francis Journals, vol. 13(4), pages 483-496.
    17. Su, Jianxi & Furman, Edward, 2017. "Multiple risk factor dependence structures: Copulas and related properties," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 109-121.
    18. Su, Jianxi & Furman, Edward, 2017. "A Form Of Multivariate Pareto Distribution With Applications To Financial Risk Measurement," ASTIN Bulletin, Cambridge University Press, vol. 47(1), pages 331-357, January.
    19. Hiroyuki Okamura & Tadashi Dohi & Kishor S. Trivedi, 2013. "Improvement of expectation–maximization algorithm for phase‐type distributions with grouped and truncated data," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 29(2), pages 141-156, March.
    20. Asimit, Alexandru V. & Furman, Edward & Vernic, Raluca, 2010. "On a multivariate Pareto distribution," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 308-316, April.
    21. Furman, Edward & Landsman, Zinoviy, 2010. "Multivariate Tweedie distributions and some related capital-at-risk analyses," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 351-361, April.
    22. Michael Kalkbrener, 2005. "An Axiomatic Approach To Capital Allocation," Mathematical Finance, Wiley Blackwell, vol. 15(3), pages 425-437, July.
    23. Joseph Kim, 2010. "Conditional Tail Moments of the Exponential Family and Its Related Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 14(2), pages 198-216.
    24. Hua, Lei, 2017. "On a bivariate copula with both upper and lower full-range tail dependence," Insurance: Mathematics and Economics, Elsevier, vol. 73(C), pages 94-104.
    25. 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.
    26. Olga Furman & Edward Furman, 2010. "On Some Layer-Based Risk Measures with Applications to Exponential Dispersion Models," Journal of Probability and Statistics, Hindawi, vol. 2010, pages 1-19, June.
    27. 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.
    28. Vernic, Raluca, 2006. "Multivariate skew-normal distributions with applications in insurance," Insurance: Mathematics and Economics, Elsevier, vol. 38(2), pages 413-426, April.
    29. Ren, Jiandong, 2010. "Recursive Formulas for Compound Phase Distributions – Univariate and Bivariate Cases," ASTIN Bulletin, Cambridge University Press, vol. 40(2), pages 615-629, November.
    30. Sarabia, José María & Gómez-Déniz, Emilio & Prieto, Faustino & Jordá, Vanesa, 2018. "Aggregation Of Dependent Risks In Mixtures Of Exponential Distributions And Extensions," ASTIN Bulletin, Cambridge University Press, vol. 48(3), pages 1079-1107, September.
    31. Fama, Eugene F. & French, Kenneth R., 1993. "Common risk factors in the returns on stocks and bonds," Journal of Financial Economics, Elsevier, vol. 33(1), pages 3-56, February.
    32. William F. Sharpe, 1964. "Capital Asset Prices: A Theory Of Market Equilibrium Under Conditions Of Risk," Journal of Finance, American Finance Association, vol. 19(3), pages 425-442, September.
    33. Sarabia, José María & Gómez-Déniz, Emilio & Prieto, Faustino & Jordá, Vanesa, 2016. "Risk aggregation in multivariate dependent Pareto distributions," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 154-163.
    34. 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.
    35. Furman, Edward & Zitikis, Ricardas, 2008. "Weighted risk capital allocations," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 263-269, October.
    36. Su, Jianxi & Hua, Lei, 2017. "A general approach to full-range tail dependence copulas," Insurance: Mathematics and Economics, Elsevier, vol. 77(C), pages 49-64.
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    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.
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    More about this item

    Keywords

    Systemic risk; Size-biased distribution; Phase-type distribution; Conditional tail expectation; Economic capital allocation;
    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

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