  In multi-state life insurance, an adequate balance between analytic
tractability, computational efficiency, and statistical flexibility is of great
importance. This might explain the popularity of Markov chain modelling, where
matrix analytic methods allow for a comprehensive treatment. Unfortunately,
Markov chain modelling is unable to capture duration effects, so this paper
presents aggregate Markov models as an alternative. Aggregate Markov models
retain most of the analytical tractability of Markov chains, yet are
non-Markovian and thus more flexible. Based on an explicit characterization of
the fundamental martingales, matrix representations of the expected accumulated
cash flows and corresponding prospective reserves are derived for
duration-dependent payments with and without incidental policyholder behaviour.
Throughout, special attention is given to a semi-Markovian case. Finally, the
methods and results are illustrated in a numerical example.
