Joseph Thomas Eghwerido, Titilola Obilade, An Optimal Investment Returns with N-Step Utility Functions, BMSA Volume 16, Bulletin of Mathematical Sciences and Applications (Volume 16) https://www.scipress.com/BMSA.16.96 Abstract: In this paper, we shall validate the optimal payoff of an investment with an N-step utility function, [6, 7], such that H* is the payoff at time N in every possible state say 2n; in an N period market setting. Negative exponential, logarithm, square root and power utility functions were considered as the market structures change according to a Markov chain. These models were used to predict the performances of some selected companies in the Nigeria Capital Market. The estimates for models design parameters p, q, p', q' correspond to halving or doubling of investment. The performance of any utility function is determined by the ratio q: q' of the probability of rising to falling as well as the ratio p: p' of the risk neutral probability measure of rising to the falling. Keywords: Capital Market, Investment Performances, Market Setting, Neutral Probability, Utility Functions