The Three-step method in a dynamic setting

A crucial issue in a dynamic framework is how risk valuations at different times are interrelated. In this regard, the notion of time consistency was widely introduced and discussed in the literature. A time-consistent dynamic valuation states that a future payoff preferred to another payoff at some future time point should already be preferred to this payoff today. This paper aims to construct a time-consistent, dynamic version of the Three-step method introduced in Deelstra et al. ((2020). ASTIN Bulletin: The Journal of the IAA, 50(3), 709–742.) for hybrid life Pure Endowment products, employing a backward iteration scheme. The backward scheme is illustrated in a dual-iteration approach using a Pure Endowment product without profit sharing. Furthermore, we explore the continuous-time limit of the backward scheme, incorporating profit-sharing into the Pure Endowment to investigate a hybrid life payoff. Our analysis demonstrates that the presence of the diversifiable component undermines the time-consistency of the dynamic three-step method. Consequently, the time-consistent price of the actuarial part shows a notable increase. To address this, and in accordance with Devolder and Lebègue ((2016). Risks, 4(4), 49.), we present a reduced time-consistent variant by decreasing the safety loads in each iterative step of the backward scheme.