If a firm rations capital its value is not being maximised. A value maximizing firm would invest in all projects with positive NPV. The firm may however want to maximize value subject to the constraint that the capital ceiling is not to be exceeded.

A linear programming method can be used to solve constrained maximization problems. The objective should be to select projects subject to the capital rationing constraint such that the sum of the projects NPVs is maximized.

Illustration

Management is faced with eight projects to invest in. The capital expenditures during the year has been rationed to Sh 500,000 and the projects have equal risk and therefore should be discounted at the firm's cost of capital of 10%.

Project Cost Project Cashflow NPV at the

t = 0(Shs) Life per year 10% cost

1 400,000 20 58,600 98,895

2 250,000 10 55,000 87,951

3 100,000 8 24,000 28,038

4 75,000 15 12,000 16,273

5 75,000 6 18,000 3,395

6 50,000 5 14,000 3,071

7 250,000 10 41,000 1,927

8 250,000 3 99,000 (3,802)

Required:

Determine the optimal investment sets.

Max Z = 98,895 X₁ + 87,951 X₂ + 28,038 X₃ + 16,273 X₃ + ... + (3,802) X₈

St 1 = 400,000 X₁ + 250,000 X₂ + 100,000 X₃ + ... + 250,000 X₈ 500,000

2 = 1 < X₁, X₂, X₃ ... X₈ > 0

The Optimal Budget:

Project Cost NPV

2 250,000 87,951

3 100,000 28,038

4 75,000 16,273

5 75,000 3,395

500,000135,657