In this paper we generalize the single-period Markowitz Mean-Variance portfolio selection problem . The Markowitz's model requires that after choosing the number of each security which constructs the portfolio in the beginning of the investment period , these numbers remain constant during and at the end of the investment period . We drop this assumption and consider an investment model in which the number of each security can vary randomly during the investment period . Indeed we consider a single-period investment with the property that the initial weight of each security is not equal to the final weight of that security . We redefine the notion of the rate of return of each security and show that the return of the investment in a cash account is not certain . We investigate some alternatives among risky securities which acts similar to cash accounts . For this we introduce the notion of free security and relate free securities to a riskless security.
