Projecting Mortality Rates Using a Markov Chain

Abstract

We present a mortality model where future stochastic changes in population-wide mortality are driven by a finite-state hierarchical Markov chain. A baseline mortality in an initial ‘Alive’ state is calculated as the average logarithm of the observed mortality rates. There are several more ‘Alive’ states and a jump to the next ‘Alive’ state leads to a change (typically, an improvement) in mortality. In order to estimate the model parameters, we minimized a weighted average quadratic distance between the observed mortality rates and expected mortality rates. A two-step estimation procedure was used, and a closed-form solution for the optimal estimates of model parameters was derived in the first step, which means that the model could be parameterized very fast and efficiently. The model was then extended to allow for age effects whereby stochastic mortality improvements also depend on age. Forecasting relies on state space augmentation and an innovations state space time series model. We show that, in terms of forecasting, our model outperforms a naïve model of static mortality within a few years. The Markov approach also permits an exact computation of mortality indices, such as the complete expectation of life and annuity present values, which are key in the life insurance and pension industries.