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.

The paper investigates a non-zero-sum differential investment and reinsurance problem between two alpha-robust, risk-averse competitive insurers under a time-consistent mean-variance criterion. The claim arrival processes for both insurers follow the classical Cram’er–Lundberg model, and the reinsurance premium is calculated using the variance premium principle. Each insurer can invest its surplus in one risk-free asset, one risky asset, and a defaultable corporate bond. The paper also considers the effect of bounded memory, which is characterized by the wealth process with delay. Using the game-theoretic approach, we derive the non-zero-sum differential alpha-robust equilibrium strategy and the corresponding value function by solving the extended Hamilton–Jacobi–Bellman equation system. In the numerical simulation section, we observe a phenomenon where the optimal strategy for investing in defaultable bonds decreases as the competitor’s risk aversion coefficient increases, provided that one’s own risk aversion coefficient remains constant. However, when there is a change in one’s own risk aversion coefficient, even if the competitor’s risk aversion changes in the opposite direction, the optimal investment strategy in defaultable bonds still decreases as one’s own risk aversion coefficient increases.