Annuity Valuation with Dependent Mortality

Abstract

Annuities are contractual guarantees, issued by insurance companies, pension plans, and government retirement systems, that offer promises to provide periodic income over the lifetime of individuals. It is well-known how to use univariate models of survivorship for valuing annuities.However, standard industry practice assumes independence of lives when valuing annuities where the promise is based on more than one life. This paper investigates the use of models of dependent mortality for determining annuity values.We discuss a broad class of parametric models using a bivariate survivorship function called a copula.Using data from a large insurance company to illustrate our methods, we calculate maximum likelihood estimates to calibrate the rnodel.The estimation results show strong positive dependence between joint lives. This statistically significant result translates into real economic significance. That is, there is an approximate five percent reduction in annuity values when dependent mortality models are used compared to the standard models that assume independence. We show that the results are robust in terms of the choice of parametric family of distribution functions. 32Annuity Valuation with Dependent Mortality

Modelling stochastic mortality for dependent lives

“…The copula approach has become a popular method of modelling the (non stochastic) bivariate survival function of the two lives of one couple. Working on the same data set that we will use, both Frees et al (1996) and Carriere (2000) present fully parametric models, using maximum likelihood, where the marginal distribution functions (Frees et al) or survival functions (Carriere) are assumed to be of Gompertz type. Frees et al (1996) use the Frank's copula and couple the two lives from the time of birth.…”

“…Working on the same data set that we will use, both Frees et al (1996) and Carriere (2000) present fully parametric models, using maximum likelihood, where the marginal distribution functions (Frees et al) or survival functions (Carriere) are assumed to be of Gompertz type. Frees et al (1996) use the Frank's copula and couple the two lives from the time of birth. Carriere (2000) on the other hand, discusses several copulas with more than one parameter (Frank, Clayton, Normal, Linear Mixing, Correlated Frailty), and couples the lives at the start of the observation period.…”

“…This paper then di¤ers from the aforementioned papers on bivariate survival models (Frees et al, 1996, Carriere, 2000, Youn and Shemyakin, 1999, Denuit et al, 2001) not only because we include stochastic mortality improvements at the marginal level, but also because, instead of assuming a speci…c copula, we select a best …tting one by following the Wang and Wells procedure for censored data. Using Wang and Wells means that we maintain the Archimedean assumption for the copula.…”

“…Formally, the construction (16) implies that the survival function of the …rst jump time , evaluated at time 0, and conditional on knowledge of the state process up to time , is…”

“…We apply our modelling and calibration procedure to a huge sample of joint survival data, belonging to a Canadian insurer, which has been used in order to discuss (non stochastic) joint mortality in Frees et al (1996), Carriere (2000), Shemyakin and Youn (2001) and Shemyakin (1999, 2001).…”