As-if-Markov reserves for reserve-dependent payments

In multistate life insurance, prospective reserves are commonly calculated as expectations conditioned only on the current state of the individual policy, rather than on the full observed past history, which is well motivated in Markov models, but is often done even when the empirical data does not show the Markov property. The resulting as-if-Markov prospective reserves then represent partially portfolio averaged values rather than individual values. This averaging effect is particularly relevant when individual policies are lapsed or modified, where it is common practice to credit the individual reserve to the policyholder, making the cashflow reserve-dependent. Such reserve dependence is normally avoided by applying the Cantelli theorem, but this does not work for as-if-Markov reserves without the Markov property. We show that, under mild technical assumptions, the as-if-Markov prospective reserves are still well defined despite the circularity in their definition, and we explain how they can be computed numerically by fixed-point iteration.