Age-Dependent Multi-Cohort Affine Mortality Model with Cohort Correlation

Yuxin Zhou, Len Patrick Garces, Yang Shen, Michael Sherris and Jonathan Ziveyi

Abstract: Continuous-time affine mortality models are useful in the analysis of age-cohort mortality rates as they yield a closed-form expression for survival curves which are consistent with the dynamics of latent factors driving mortality and are well-suited for finance and insurance applications. We extend and improve these mortality models by introducing age dependence of mortality rates and correlation between cohorts. We propose and compare two classes of age-dependent mortality models, namely the age-dependent coefficient model and the age-dependent factor model. Specifically, we assess both Gaussian and CIR-type models for each category of age-dependent models. Both categories of age-dependent models involve age and calendar time, which in turn specifies the cohort. Thus, our models admit an analytical form for the instantaneous correlation between mortality rates of different cohorts. Moreover, we propose two improvements to the parameter estimation process. First, to improve the estimation of cohort correlations, we regularise the parameter estimation by adding a penalty term which penalises larger differences between empirical and estimated correlations. Second, we develop and assess a method to include incomplete cohorts into the Kalman filtering algorithm for parameter estimation. We calibrate the mortality models to data from multiple countries which include Australia, Denmark, UK, and USA to assess and compare in-sample fit and forecasting performance. By incorporating age dependence and using incomplete cohort mortality data, we improve the goodness of fit and produce more reasonable out-of-sample forecasts of survival probabilities. We also show that regularisation produces more realistic correlations between cohorts for varying age differences. Our results show that, under most circumstances, the correlation between cohorts decreases as the age difference increases.

Keywords: Mortality model, Age-dependent, Multi-cohort, Cohort correlation, Incomplete cohort, Affine, Regularisation