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We consider an expected utility maximization problem where the utilityfunction is not necessarily concave and the time horizon is uncertain. Weestablish a necessary and sufficient condition for the optimality for generalnon-concave utility function in a complete financial market. We show that thegeneral concavification approach of the utility function to deal withnon-concavity, while being still applicable when the time horizon is a stoppingtime with respect to the financial market filtration, leads to sub-optimalitywhen the time horizon is independent of the financial risk, and hence can notbe directly applied. For the latter case, we suggest a recursive procedurewhich is based on the dynamic programming principle. We illustrate our findingsby carrying out a multi-period numerical analysis for optimal investmentproblem under a convex option compensation scheme with random time horizon. Weobserve that the distribution of the non-concave portfolio in both certain anduncertain random time horizon is right-skewed with a long right tail,indicating that the investor expects frequent small losses and a few largegains from the investment. While the (certain) average time horizon portfolioat a premature stopping date is unimodal, the random time horizon portfolio ismultimodal distributed which provides the investor a certain flexibility ofswitching between the local maximizers, depending on the market performance.The multimodal structure with multiple peaks of different heights can beexplained by the concavification procedure, whereas the distribution of thetime horizon has significant impact on the amplitude between the modes.

Non-concave expected utility optimization with uncertain time horizon