Capital budgeting problem

From Glossary

Jump to: navigation, search

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.)

Portfolio selection problembutton.png

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)].

Personal tools