Probability Distributions of Multi-States Insurance Models under Semi-Markov Assumptions

In this paper we develop integral and differential backward equation systems for the state-wise cumulative distribution functions of the discounted future cash-flow of multi-states life and health insurance policies, assuming that the state process of the insured is semi-Markovian. We derive equation systems in two different ways: First, we apply the inversion method on the state-wise moment generating function. Second, we apply a transformation trick on the state-wise cumulative distribution functions. The results are illustrated with applications to Value-at-Risk (VaR) and Tail-Value-at-Risk (TVaR) calculations.