We shall propose a new computational scheme for the evaluation of the optimal portfolio for investment.Our method is based on an extension of the asymptotic expansion approach which has been recently developed for pricing problems of the contingent claims ' analysis by Kunitomo-Takahashi(1992, 1995, 1998, 2001), Yoshida(1992), Takahashi(1995,1999), Takahashi and Yoshida(2001). In particular, we will explicitly derive a formula of the optimal portfolio associated with maximizing utility from terminal wealth in a nancial market with Markovian coe cients,and give a numerical example for a power utility function.