Faculty of Actuarial Science & Insurance Seminar with Sabrina Mulinacci (University of Bologna)

Abstract

In this talk we analyze some alternative models for joint residual lifetimes allowing for the simultaneous death of the involved individuals.

Bivariate copula functions have been widely used to model the dependence structure between the residual lifetimes of the two individuals in a couple. However, most commonly used copulas are absolutely continuous and do not allow for the case of a simultaneous death. In the first part of the talk, we consider the Extended Marshall-Olkin (EMO) model, see Gobbi et al. (2019), which is based on the combination of two approaches: the absolutely continuous copula approach, where the copula is used to capture dependencies due to environmental factors shared by the two lives,
and the classical Marshall-Olkin model, where the association is given by accounting for a shock related to a fatal event causing the simultaneous death of the two lives. Nevertheless, there are many practical situations in which the effect of a shock is not immediate, but the fatal event is registered with some delay with respect to the shock occurrence. Staring from the Marshall-Olkin model with implicit common shocks studied in Ryu (1993), we construct a model that generalizes the EMO one, allowing for a possible delay in the effect of the common shock in the two lives in a couple: we call it “Ryu-type Extended Marshall-Olkin (REMO) model” (see Gobbi et a., 2021).  Relevant properties of EMO and REMO models are analyzed and the behavior of the induced mortality intensities studied. Finally, both models are applied to a sample of censored residual lifetimes of couples of insureds extracted from a dataset of annuities contracts.

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