On age difference in joint lifetime modelling with life insurance annuity applications

Abstract

Insurance and annuity products covering several lives require the modelling of the joint distribution of future lifetimes. In the interest of simplifying calculations, it is common in practice to assume that the future lifetimes among a group of people are independent. However, extensive research over the past decades suggests otherwise. In this paper, a copula approach is used to model the dependence between lifetimes within a married couple using data from a large Canadian insurance company. As a novelty, the age difference and the gender of the elder partner are introduced as an argument of the dependence parameter. Maximum likelihood techniques are thus implemented for the parameter estimation. Not only do the results make clear that the correlation decreases with age difference, but also the dependence between the lifetimes is higher when husband is older than wife. A goodness-of-fit procedure is applied in order to assess the validity of the model. Finally, considering several products available on the life insurance market, the paper concludes with practical illustrations.Keywords: Dependent lifetimes, Copula and dependence, Goodness-of-fit, Maximum likelihood estimator, Life insurance. the increased mortality of widowers, often called the broken heart syndrome. Many contributions have shown that there could be a significant difference between risk-related quantities, such as risk premiums, evaluated according to dependence or independence assumptions. Denuit and Cornet [9] have measured the effect of lifetime dependencies on the present value of a widow pension benefit. Based on the data collected in cemeteries, not only do their estimation results confirm that the mortality risk depends on the marital status, but also show that the amounts of premium are reduced approximately by 10 per cent compared to model which assumes independence. According to data from a large Canadian insurance company, Frees et al. [11] have demonstrated that there is a strong positive dependence between joint

Stochastic Mortality Modelling for Dependent Coupled Lives

“…Additional dependence between lifetimes further to the contagion effects resulting directly from the loss is therefore generated through implementation of their model. Although dependence due to unobserved risk factors is not considered by Dufresne et al (2018), incorporation of age difference and sex of the elder spouse in definition of the level of association between coupled lifetimes in a copula model, supports suggestion of asymmetric dependence. Jevtić and Hurd (2017) introduce an alternative to copula dependence in the credit risk environment through definition of a probabilistic mechanism describing the influence the primary death has on the bereaved spouse.…”

“…In the case of widowers, the pattern of changing mortality differs slightly from the general findings of the investigation, 13.7% of widowers at risk died within the first six months following the bereavement whilst just 5.9% died in the second, a difference in mortality found to be significant at the 1% level. Frees et al (1996), Carriere (2000), Youn and Shemyakin (1999), Shemyakin and Youn (2006), Spreeuw and Wang (2008), Luciano et al (2016) and Dufresne et al (2018) are amongst a number of studies making use of data from a large Canadian insurer. Observations include increased mortality for widows and widowers in comparison to non-widowed lives, maximum mortality rates among individuals having lost their partner less than one year ago and the implication of a greater influence of broken-heart syndrome on men rather than women.…”

Joint Life Insurance Pricing Using Extended Marshall–olkin Models

“…As done in Dufresne et al (2018) for the same data set, we will apply the two-stage parametric method for censored data introduced in Shih and Louis (1995).…”

“…In actuarial literature many families of distributions have been proposed and studied to fit the distributional properties of residual lifetimes. Taking into account the data set on which we want to test the EMO model, we assume that marginal residual lifetimes follow a Gompertz law, taking into account the fact that in Frees et al (1996) and in Carriere (2000) it was proved that this distribution performed a very good fit in this data set (the same choice is done in Dufresne et al (2018), and in a generalized approach in Luciano et al (2008), Ji et al (2011) and Luciano et al (2016)). More precisely, we assume that residual lifetimes survival distributions from ages x m and x f are given bȳ…”

“…The copula approach was firstly adopted in Frees et al (1996), where a data set on annuities contracts on the joint life of couples provided by a Canadian insurer is analyzed. Many papers have been then devoted to the use of copula functions to model the dependence among the lifetimes in a couple and most of them focus on the absolutely continuous copulas (mainly Archimedean copulas): see Youn and Shemyakin (1999), Carriere (2000), Denuit et al (2001), Spreeuw (2006), Shemyakin and Youn (2006), Luciano et al (2008Luciano et al ( , 2016, Dufresne et al (2018). As for the brokenheart syndrome, in addition to the multi-state Markov chain method already mentioned, a frailty-based approach has been considered in Gouriéroux and Lu (2015): this tool has been further generalized and estimated on a French reinsurer database in Lu (2017).…”