# 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.)