Portfolio Selection under Condition of Variable Weights

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 as-sump- tion and consider an investment model in which the number of each security can vary randomly during the in-vestment 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 securi-ties which acts similar to cash accounts. For this we introduce the notion of free security and relate free securities to a riskless se- curity.