Modelling stochastic mortality for dependent lives

Abstract

Stochastic mortality, i.e. modelling death arrival via a jump process with stochastic intensity, is gaining an increasing reputation as a way to represent mortality risk. This paper is a first attempt to model the mortality risk of couples of individuals, according to the stochastic intensity approach. Dependence between the survival times of the members of a couple is captured by an Archimedean copula. We also provide a methodology for fitting the joint survival function by working separately on the (analytical) marginals and on the (analytical) copula. First, we provide a sample-based calibration for the intensity, using a time-homogeneous, non mean-reverting, affine process: this gives the marginal survival functions. Then we calibrate and select the best fit copula according to the Wang and Wells [Wang, W., Wells, M.T., 2000b. Model selection and semiparametric inference for bivariate failure-time data. J. Amer. Statis. Assoc. 95, 62–72] methodology for censored data. By coupling the calibrated marginals with the best fit copula, we obtain a joint survival function, which incorporates the stochastic nature of mortality improvements. We apply the methodology to a well known insurance data set, using a sample generation. The best fit copula turns out to be one listed in [Nelsen, R.B., 2006. An Introduction to Copulas, Second ed. In: Springer Series], which implies not only positive dependence, but dependence increasing with age.