PORTFOLIO SELECTION PROBLEM

Portfolio selection problem - In its elementary form, this is the same as the capital budgeting problem , except that the objective is to minimize the risk, rather than maximize expected return. Let x_j be the percent of capital invested in the j-th opportunity (e.g., stock or bond), so x must satisfy x >= 0 and Sum_j(x_j)=1. Let v_j be the expected return per unit of investment in the j-th opportunity, so that vx is the sum total rate of return per unit of capital invested. It is required to have a lower limit on this rate: vx >= b (where b is between Min(v_j) and Max(v_j)). Subject to this rate of return constraint, the objective is the quadratic form, x'Qx, where Q is the variance-covariance matrix associated with the investments (i.e., if the actual return rate is V_j, then Q(i,j) = E[(V_i - v_i)(V_j - v_j)].