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

Edward Furman, Yisub Kye and Jianxi Su

Insurance: Mathematics and Economics, 2021, vol. 96, issue C, 153-167

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.

Keywords: Systemic risk; Size-biased distribution; Phase-type distribution; Conditional tail expectation; Economic capital allocation (search for similar items in EconPapers)
JEL-codes: C02 C46 (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (9)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:insuma:v:96:y:2021:i:c:p:153-167

DOI: 10.1016/j.insmatheco.2020.11.007

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