Abstract
We consider a non-homogeneous continuous time Markov chain model for Long-Term Care with five states: the autonomous state, three dependent states of light, moderate and severe dependence levels and the death state. For a general approach, we allow for non null intensities for all the returns from higher dependence levels to all lesser dependencies in the multi-state model. Using data from the 2015 Portuguese National Network of Continuous Care database, as the main research contribution of this paper, we propose a method to calibrate transition intensities with the one step transition probabilities estimated from data. This allows us to use non-homogeneous continuous time Markov chains for modeling Long-Term Care. We solve numerically the Kolmogorov forward differential equations in order to obtain continuous time transition probabilities. We assess the quality of the calibration using the Portuguese life expectancies. Based on reasonable monthly costs for each dependence state we compute, by Monte Carlo simulation, trajectories of the Markov chain process and derive relevant information for model validation and premium calculation.
Highlights
Population ageing is a phenomenon which, undoubtedly, is inevitable in the future, in all regions of the world, see (OECD 2019)
In (Rolski et al 1999, p. 349) we find a construction of the non-homogeneous continuous time Markov chain from an initial distribution and an intensity matrix, allowing for simulation
We aim to relax the assumption of discrete model, allowing to develop a continuous-time Markov model framework, in order to further develop well known pricing and reserving techniques for LTC insurance products, as well as to provide information to the Portuguese National Network for Long-Term Care that allows for cost estimation with this public service
Summary
Population ageing is a phenomenon which, undoubtedly, is inevitable in the future, in all regions of the world, see (OECD 2019). Forecasts point to a severe ageing of the world population and Portugal is not an exception as pointed out, e.g., in (OECD 2013) This social/economic problem of increasing proportions carries many difficulties to solve, namely the dependence of elderly and convenient provision of Long-Term Care (LTC)—. That is, the health and well-being support needed in the later stages of life—so it becomes imperative to give special attention to this worldwide problem This is an issue that is on the agenda of many countries, especially the developed ones, where the ageing phenomenon, and the elderly dependence, is more evident. Some studies have already been published, using real data, such as (Guillen 2008; Fuino and Wagner 2018) In some of these countries the insurance market is already providing LTC products.
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