Modelling stochastic mortality for dependent lives

Abstract

Stochastic mortality, i.e. modelling death arrival via a jump process with stochastic intensity, is gaining increasing reputation as a way to represent mortality risk. This paper represents a …rst attempt to model the mortality risk of couples of individuals, according to the stochastic intensity approach. We extend to couples the Cox processes set up, namely the idea that mortality is driven by a jump process whose intensity is itself a stochastic process, proper of a particular generation within each gender. Dependence between the survival times of the members of a couple is captured by an Archimedean copula. We also provide a methodology for …tting the joint survival function by working separately on the (analytical) copula and the (analytical) margins. First, we calibrate and select the best …t copula according to the methodology of Wang and Wells (2000b) for censored data. Then, we provide a sample-based calibration for the intensity, using a time-homogeneous, non mean-reverting, a¢ ne process: this gives the marginal survival functions. By coupling the best …t copula with the calibrated margins we obtain a joint survival function which incorporates the stochastic nature of mortality improvements. Several measures of time dependent association can be computed out of it.We apply the methodology to a well known insurance dataset, using a sample generation. The best …t copula turns out to be a Nelsen one, which implies not only positive dependency, but dependency increasing with age.JEL Classi…cation: G22

Spouses’ Dependence across Generations and Pricing Impact on Reversionary Annuities

“…Thanks to the fit exploration provided in [6] on the same dataset, we consider the Archimedean copulas, listed in Table 1. Table 1.…”

“…It is reasonable to assume that persons with ages of birth close to each other belong to the same generation. As in [6], we define a generation as the set of all individuals born in a fourteen-year time interval. We set the age difference between male and female of the same couple equal to three years, as this is the average age difference between spouses in the whole dataset.…”

“…For instance, the male members of the old generation may start to be observed at every age between 75 and 94. For notational convenience and according to [6], we will consider as the initial age of each generation the smallest possible entry age, namely, x = 75 for the old male, x = 61 for the young male, y = 72 for the old female, y = 58 for the young female. 2 We did fit more common 2P families, such as Student's copula and several families in [22], to the data, as well, but the fits obtained were not as good as the ones resulting from the three types specified above.…”

Archimedean Copulas Derived from Morgenstern Utility Functions

“…Some properties of the copula generated are set out in [33], whereas the family is also involved in a study on mortality of coupled lives in [23], who named it the "Special Copula". Both these contributions give a representation in terms of the inverse of the derivative which is…”

Profitability of Pension Contributions: Evidence from Real-Life Employment Biographies

“…Official mortality statistics also do not reveal the interdependencies in the remaining lifetimes of spouses. According to Luciano et al (2008), when one of the spouses dies, the partner's probability of dying rises. Then our IRR estimates for married insurants are too high as expected returns from widow(er) pensions are overvalued.…”