# Capital budgeting problem

### From Glossary

In its elementary form there is a fixed amount of capital, say , that can be allocated to any of investments. Each investment has a minimum level, say , and a maximum level, say . The expected return on investment is a function, , where is the level of the -th investment opportunity (). Risk is measured by a standard deviation from the expected return, say . The problem is to maximize total expected return, subject to a *budget constraint*: , and a *risk constraint*: , where and are parameters. The returns on the investments could be correlated. Then, if is the variance-covariance matrix, the risk constraint is quadratic: . (Also see the 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 be the percent of capital invested in the j-th opportunity (e.g., stock or bond), so must satisfy and Let be the expected return per unit of investment in the j-th opportunity, so that is the sum total rate of return per unit of capital invested. It is required to have a lower limit on this rate: (where is between and ). Subject to this rate of return constraint, the objective is the quadratic form, where is the variance-covariance matrix associated with the investments (i.e., if the actual return rate is then