Modelling USA Age-Cohort Mortality: A Comparison of Multi-Factor Affine Mortality Models

Abstract

Affine mortality models are well suited for theoretical and practical application in pricing and risk management of mortality risk. They produce consistent, closed-form stochastic survival curves allowing for the efficient valuation of mortality-linked claims. We model USA age-cohort mortality data using five multi-factor affine mortality models. We focus on three-factor models and compare four Gaussian models along with a model based on the Cox–Ingersoll–Ross (CIR) process, allowing for Gamma-distributed mortality rates. We compare and assess the Gaussian Arbitrage-Free Nelson–Siegel (AFNS) mortality model, which incorporates level, slope and curvature factors, and the canonical Gaussian factor model, both with and without correlations in the factor dynamics. We show that for USA mortality data, the probability of negative mortality rates in the Gaussian models is small. Models are estimated using discrete time versions of the models with age-cohort data capturing variability in cohort mortality curves. Poisson variation in mortality data is included in the model estimation using the Kalman filter through the measurement equation. We consider models incorporating factor dependence to capture the effects of age-dependence in the mortality curves. The analysis demonstrates that the Gaussian independent-factor AFNS model performs well compared to the other affine models in explaining and forecasting USA age-cohort mortality data.

AffineMortality: An R package for estimation, analysis, and projection of affine mortality models

“…Here, x is fixed and denotes the smallest age in the age-range of interest (for example, equal to 50 in Ungolo et al, 2023 andHuang et al, 2022). The dataset for the analysis is a matrix of dimension N × K, where N is the number of ages in the age-range of interest and K is the number of calendar years for the analysis.…”

“…The use of the average forces of mortality yields smoother data, which renders the estimation process more stable. This is the approach adopted within the interest rate literature (see Christensen et al, 2011), as well as in the analysis of affine mortality models (Blackburn & Sherris, 2013;Huang et al, 2022 andRegis, 2019).…”

“…This paper describes the R package AffineMortality, which supports an extensive analysis of continuous time affine mortality models in the spirit of Blackburn & Sherris (2013), Huang et al (2022), and Ungolo et al (2023). More precisely, the package estimates the parameters of these models using mortality data collected at discrete time points and ages.…”

Should Selection of the Optimum Stochastic Mortality Model Be Based on the Original or the Logarithmic Scale of the Mortality Rate?

“…Here, most extensions of the LC model define the reference mortality model assuming a Gaussian error structure of log mortality rates (Chang and Shi 2022;Gao and Shi 2021;Li and Lu 2017;Li and Shi 2021;SriDaran et al 2022) or a Poisson distribution of deaths (Barigou et al 2021;Chen and Millossovich 2018;Enchev et al 2017;Hunt and Blake 2014;Li 2013;Li et al 2016Pitt et al 2018;Wong et al 2020;Yang et al 2016). A less common option for the reference mortality model is to assume a binomial distribution of annual death probabilities (Atance et al 2020) or gamma distribution for mortality rates (Huang et al 2022).…”

“…As Atance et al (2020) stress, there is no single criterion for evaluating the goodness-of-fit and the prediction accuracy of stochastic mortality models. Selection criteria frequently rely on measures based on squared errors (Chang and Shi 2022;Enchev et al 2017;Gao and Shi 2021;Li and Lu 2017;Li and Shi 2021), absolute errors (Li et al 2016, maximum likelihood (Pitt et al 2018;Yang et al 2016) or a combination of these measures (Atance et al 2020;Chen and Millossovich 2018;Huang et al 2022;Li 2013;Wong et al 2020). Additionally, even the same selection criteria measures are often defined based on either mortality rate predictions (estimates) (Atance et al 2020;Chen and Millossovich 2018) or log mortality rate predictions (estimates) (Chang and Shi 2022;Enchev et al 2017;Gao and Shi 2021;Li and Lu 2017;Li and Shi 2021;Li and Lee 2005;Wong et al 2020).…”