# 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 $LaTeX: x_j$ be the percent of capital invested in the j-th opportunity (e.g., stock or bond), so $LaTeX: x$ must satisfy $LaTeX: \textstyle x \ge 0$ and $LaTeX: \textstyle \sum_j x_j = 1.$ Let $LaTeX: v_j$ be the expected return per unit of investment in the j-th opportunity, so that $LaTeX: vx$ is the sum total rate of return per unit of capital invested. It is required to have a lower limit on this rate: $LaTeX: \textstyle vx \ge b$ (where $LaTeX: b$ is between $LaTeX: \min v_j$ and $LaTeX: \max v_j$). Subject to this rate of return constraint, the objective is the quadratic form, $LaTeX: x^TQx,$ where $LaTeX: Q$ is the variance-covariance matrix associated with the investments (i.e., if the actual return rate is $LaTeX: V_j,$ then $LaTeX: \textstyle Q(i,j) = E[(V_i - v_i)(V_j - v_j)].$