A dynamic bivariate common shock model with cumulative effect and its actuarial application

Abstract

Standard actuarial theory of multiple life insurance traditionally postulates independence for the remaining lifetimes mainly due to computational convenience rather than realism. In this paper, we propose a general common shock model for modelling dependent coupled lives and apply it to a life insurance model. In the proposed shock model, we consider not only simultaneous deaths of the coupled members due to a single shock (e.g. a critical accident), but also cumulative effect in the mortality rate when they survive shocks. Under the model, we derive a bivariate lifetime distribution and its marginal distributions in closed forms. We study the bivariate ageing property, dependence structure and the dependence orderings of the lifetime distribution. Based on it, we investigate the influence of dependence on the pricings of insurance policies involving multiple lives which are subject to common shocks. Furthermore, we discuss relevant useful stochastic bounds.

Keywords:

• Common shock model
• shot noise process
• joint life
• stochastic dependence
• life annuities

Acknowledgements

The authors greatly appreciate the reviewer’s insightful comments and constructive suggestions, which have improved the presentation of this paper substantially.

Additional information

Funding

This work was supported by Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology [grant number 2009-0093827]; the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) [grant number 2016R1A2B2014211]. The work of the first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education [grant number 2-2017-1659-001-1].