A method for calculating the implied no-recovery three-state transition matrix using observable population mortality incidence and disability prevalence rates among the elderly

The most accurate estimation of transition probabilities for a multi-state model of health status requires longitudinal data. However, for many countries such data are usually not available. Instead, population level mortality incidence and disability prevalence rates are often all that can be accessed. In this paper, for a three-state no-recovery model (with states healthy, disabled, dead), using simple mathematical derivations, we propose a framework to estimate the age- and gender-specific boundaries within which each of the transition probabilities should fall. We then provide two methods for estimating unique transition probabilities—a least squares procedure and a method based on the ‘extra mortality’ factor proposed by Rickayzen and Walsh (Br Actuarial J 8(2):341–393, 2002, https://doi.org/10.1017/s1357321700003755). We also show the acceptable range for the ‘extra mortality’ factor given the mortality and disability data. Furthermore, we provide a critique of the method proposed by Van der Gaag et al. (Demogr Res 32:75, 2015), as their estimates can fall outside the acceptable boundaries. Finally, we estimate life and health expectancies, as well as premium rates for a life care annuity and a disability annuity using our derived transition probabilities.