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
Most publications on modeling insurance contracts on two lives, assuming dependence of the two lifetimes involved, focus on the time of inception of the contract. The dependence between the lifetimes is usually modeled through a copula and the effect of this dependence on the pricing of a joint life policy is measured. This paper investigates the effect of association at the outset on the mortality in the future. The conditional law of mortality of an individual, given his survival and given the life status of the partner is derived. The conditional joint survival distribution of a couple at any duration, given that the two lives are then alive, is also derived. We analyze how the degree of dependence between the two members of a couple varies throughout the duration of a contract. We will do that for (mainly Archimedean) copula models, with one parameter for the degree of dependence. The conditional distributions hence derived provide the basis for the calculation of prospective provisions.
Citation: Spreeuw, J. & Owadally, M. I (2013). Investigating the broken-heart effect: a model for short-term dependence between the remaining lifetimes of joint lives. Annals of Actuarial Science, 7(2), pp. 236-257. doi: 10.1017/S1748499512000292 This is the accepted version of the paper.This version of the publication may differ from the final published version. Permanent June 2012 AbstractWe analyse the mortality of couples by fitting a multiple state model to a large insurance data set. We find evidence that mortality rates increase after the death of a partner and, in addition, that this phenomenon diminishes over time. This is popularly known as a "broken-heart" effect and we find that it affects widowers more than widows. Remaining lifetimes of joint lives therefore exhibit short-term dependence. We carry out numerical work involving the pricing and valuation of typical contingent assurance contracts and of a joint life and survivor annuity. If insurers ignore dependence, or mis-specify it as long-term dependence, then significant mis-pricing and inappropriate provisioning can result. Detailed numerical results are presented.
This paper studies the dependence between coupled lives, i.e., the spouses' dependence, across different generations, and its effects on prices of reversionary annuities in the presence of longevity risk. Longevity risk is represented via a stochastic mortality intensity. We find that a generation-based model is important, since spouses' dependence decreases when passing from older generations to younger generations. The independence assumption produces quantifiable mispricing of reversionary annuities, with different effects on different generations. The research is conducted using a well-known dataset of double life contracts.
The (additive) generator of an Archimedean copula is a strictly decreasing and convex function, while Morgenstern utility functions (applying to risk aversion decision makers) are nondecreasing and concave. In this presentation, relationships between generators and utility functions are established. For some well known Archimedean copula families, links between the generator and the corresponding utility function are demonstrated. Some new copula families are derived from classes of utility functions which appeared in the literature, and their properties are discussed. It is shown how dependence properties of an Archimedean copula translate into properties of the utility function from which they are constructed.
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