This paper presents a novel approach in the Monte Carlo code TRIPOLI-4® for estimating the asymptotic reactor period by solving the α-eigenvalue equation, accounting for contributions from both neutrons and precursors. The proposed method incorporates a tallying technique to update the α eigenvalue during a modified power iteration by a root-finding procedure, which effectively addresses some of the shortcomings of the legacy α-K algorithm, particularly regarding the constraints imposed by the initial guess and the rate of convergence. Three different variants of the new procedure have been implemented: the standard version introduced by Josey and Brown for MCNP, a block-average approach that updates the α eigenvalue every M generations to reduce the sensitivity to stochastic variations, and a third version where a change of variable is applied in order to quench the bias affecting the final estimate of α. These algorithms have been extensively tested against the legacy α-K method using Cullen’s set of spherical configurations and the CROCUS benchmark.