Correlated age-specific mortality model: an application to annuity portfolio management
In this paper, the authors discuss the modelling of the dynamics of age-specific incremental mortality using a stochastic process with the objective of incorporating mortality correlations across ages. Specifically, they model the drift rate as the average annual improvement rate of a group time trend for all ages and consider both normal and non-normal distributions for the distribution of residuals. Some key features such as skewness, heavy-tailedness, jumps are captured in the non-normal residual distributions. To incorporate the inter-age mortality dependence, the authors employ a one-factor copula model with six distributions. Applications of the proposed models to pricing and quantifying risks of three annuity portfolios are provided. Specifically, the authors propose approximations to the change in the value of an annuity portfolio with respect to a proportional or constant change in the force of mortality. They also consider the estimation of value at risk (VaR) and some summary statistics, say the mean, variance, skewness, kurtosis, of the loss distribution for each of the three annuity portfolios. The specification of the proposed age-specific mortality model is presented in Section 2. Section 2.1 introduces three approaches for the estimation of annual drift rates of the logarithmic forces of mortality. Section 2.2 discusses several non-normal distributions, say a jump-diffusion distribution, a Student's t distribution, a normal inverse Gaussian distribution and a variance Gamma distribution, for the residuals. Section 3 presents the one-factor copula model, where the latent variables for different ages depend on one common factor and an idiosyncratic factor independent of the common factor. A copula function is then used to modelling the dependency of the latent variables for different ages. Six pairs of distributions for the common factor and the idiosyncratic factor are considered. In Section 4, the models are fitted to mortality data for both genders of the United States and Japan from the Human Mortality Database. Three annuity portfolios based on the barbell strategy, the bullet strategy and the ladder strategy are constructed in Section 5. These strategies are developed by investing different proportions of capitals in bonds with different strategies. The duration and convexity of an annuity product are discussed in Section 5.1. The three annuity portfolios are constructed in Section 5.2, where the relevant parameters are estimated using mortality data. The designing, pricing and risk management of annuity products are discussed in Section 6. Approximations to static changes in portfolio values with respect to a proportional or constant change in the force of mortality are provided in Section 6.1. The estimation of the VaR for the longevity risks of the three annuity portfolios are considered in Section 6.2. Some summary statistics for the loss distributions of the three annuity portfolios are presented. From the numerical results, the authors highlight the significance of the residual distributions and the inter-age dependence for estimating the longevity risks of the three annuity portfolios.
scientific article; zbMATH DE number 7457619
scientific article