Aggregate Markov Models in Life Insurance: Properties and Valuation

In multi-state life insurance, the random pattern of states of the insured is usually assumed to follow a time-inhomogeneous Markov jump process on a finite state space. In the context of phase-type distributions, this setting leads to sojourn times in the different states that follow one-dimensional inhomogeneous phase-type distributions.

In this talk, we introduce a so-called aggregated Markov chain model in life insurance, where we assign a number of micro states to each biometric macro state, leading to sojourn times in macro states following inhomogeneous phase-type distributions of general dimension. Although this model in general lead to path dependencies in the states of the insured, we are able to take advantage of the properties of inhomogeneous phase-type distributions and derive explicit and tractable expressions for the distribution of jump times and transitions based on probabilistic sample path arguments. These properties are then used to derive expressions for expected life insurance cash flows in a setup where payments are allowed to depend on the duration in the different states.

From our general results, we identify an important special case leading to sojourn time distributions being independent of past jump times and transition. We show that this implies a specific semi-Markovian structure on the states of the insured, and we then link this special case to the classic semi-Markovian life insurance models known from existing literature. We end the talk by presenting a numerical example of the results, which serves to illustrate applicability of our results to examples known from existing literature and actuarial practice.

Find the Q&A here: Q&A on 'Mortality Models'