Capital budgeting problem

In its elementary form there is a fixed amount of capital, say $LaTeX: C$, that can be allocated to any of $LaTeX: n$ investments. Each investment has a minimum level, say $LaTeX: L$, and a maximum level, say $LaTeX: U$. The expected return on investment is a function, $LaTeX: v_j(x_j)$, where $LaTeX: x_j$ is the level of the $LaTeX: j$-th investment opportunity ($LaTeX: L_j \le x_j \le U_j$). Risk is measured by a standard deviation from the expected return, say $LaTeX: s_j(x_j)$. The problem is to maximize total expected return, subject to a budget constraint: $LaTeX: \sum_j x_j \le C$, and a risk constraint: $LaTeX: \sum_j v_j(x_j) + a_j s_j(x_j)} \le b$, where $LaTeX: a_j$ and $LaTeX: b$ are parameters. The returns on the investments could be correlated. Then, if $LaTeX: Q$ is the variance-covariance matrix, the risk constraint is quadratic: $LaTeX: v^Tx + x^T Q x \le b$. (Also see the 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 $LaTeX: \textstyle Q(i,j) = E[(V_i - v_i)(V_j - v_j)].$