Bivariate Compound Tail Probabilities and Their Bounds for the Assessment of Required Capital

27 Pages Posted: 22 Sep 2022

See all articles by Ang Li

Ang Li

Western University, Canada

Jiandong Ren

University of Western Ontario

Ricardas Zitikis

Western University

Date Written: September 6, 2022

Abstract

To assess the required capital for insurance companies, which usually have several business lines, estimates of the joint tail probabilities of aggregate losses are needed. Under the collective risk model, these aggregate losses are sums of random number of random variables, whose distributions and thus, in turn, regulatory risk measures such as the value at risk (VaR), have been notoriously difficult to derive. Hence, many actuarial scholars have worked on deriving upper bounds for the tail probabilities whose relative simplicity allowed them to estimate, for example, the VaR at various probability levels with practically acceptable precisions. In the case of several business lines, this requires estimates for the probabilities of multivariate compound sums, a topic that, despite its importance, has recently been somewhat neglected in the actuarial literature. In this paper we rejuvenate this line of research and develop estimates for the tail probabilities in the case of two business lines. In particular, we derive upper bounds for the joint tail probabilities when the claim severities may, or may not, have finite moment generating functions. We assess the tightness of the bounds using numerical examples.

Keywords: bivariate distribution, compound sum, required capital, joint tail probability, value at risk, estimates

Suggested Citation

Li, Ang and Ren, Jiandong and Zitikis, Ricardas, Bivariate Compound Tail Probabilities and Their Bounds for the Assessment of Required Capital (September 6, 2022). Available at SSRN: https://ssrn.com/abstract=4211824 or http://dx.doi.org/10.2139/ssrn.4211824

Ang Li

Western University, Canada ( email )

Jiandong Ren

University of Western Ontario ( email )

1151 Richmond Street
Suite 2
London, Ontario N6A 5B8
Canada

Ricardas Zitikis (Contact Author)

Western University ( email )

1151 Richmond Street
Suite 2
London, Ontario N6A 5B8
Canada

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