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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.

Suggested Citation

  • 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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    2. Mercè Claramunt, M. & Lefèvre, Claude & Loisel, Stéphane & Montesinos, Pierre, 2022. "Basis risk management and randomly scaled uncertainty," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 123-139.
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    4. 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.
    5. Gribkova, N.V. & Su, J. & Zitikis, R., 2022. "Inference for the tail conditional allocation: Large sample properties, insurance risk assessment, and compound sums of concomitants," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 199-222.
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    7. Denuit, Michel & Ortega-Jimenez, Patricia & Robert, Christian Y., 2024. "No-sabotage under conditional mean risk sharing of dependent-by-mixture insurance losses," LIDAM Discussion Papers ISBA 2024019, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    8. Marri, Fouad & Moutanabbir, Khouzeima, 2022. "Risk aggregation and capital allocation using a new generalized Archimedean copula," Insurance: Mathematics and Economics, Elsevier, vol. 102(C), pages 75-90.
    9. 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.

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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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