CHAPTER 14

LONG-RUN INVESTMENT DECISIONS: CAPITAL BUDGETING

1. A firm's marginal cost of capital (i) in percentage terms is a linear function of the total amount it chooses to borrow (K) in millions of dollars during the current time period. The function is i = 5 + 0.8K. The firm is considering six projects that will have to be financed entirely by borrowing. The amount the firm must borrow for each project and the expected rates of return are listed below:

Project                   Amount borrowed (K)                        Rate of return

   A                                       $4.2                                              22 percent

   B                                          5.8                                             18

   C                                          3.5                                             16

   D                                          1.5                                             14

   E                                           5.3                                             12

   F                                           1.7                                             10

Which projects should the firm undertake, if any, and what will the firm's marginal cost of capital be if it borrows the optimal amount?

Solution:

Project                            Total amount borrowed                  Cost of capital

     A                                            $4.2                                            8.36 percent

     B                                             10.0                                           13.00

     C                                             13.5                                           15.80

     D                                            15.0                                           17.00

     E                                             20.3                                           21.24

     F                                             22.0                                           22.60

The firm should borrow a total of $13.5 million and carry out projects A, B, and C. The firm's marginal cost of capital will be 15.8 percent.

2. A firm's marginal cost of capital (i) in percentage terms is a linear function of the total amount it chooses to borrow (K) in millions of dollars during the current time period. The function is i = 4.5 + 0.5K. The firm is considering six projects that will have to be financed entirely by borrowing. The amount the firm must borrow for each project and the expected rates of return are listed below:

Project                   Amount borrowed (K)                      Rate of return

     A                                  $2.8                                             16 percent

     B                                     1.9                                            15

     C                                     3.3                                            12

     D                                    1.7                                             11

     E                                     5.3                                             10

     F                                     6.0                                                9

Which projects should the firm undertake, if any, and what will the firm's marginal cost of capital be if it borrows the optimal amount?

Solution:

Project                        Total amount borrowed                  Cost of capital

     A                                          $2.8                                          5.90 percent

     B                                            4.7                                           6.85

     C                                            8.0                                           8.50

     D                                            9.7                                           9.35

     E                                          15.0                                          12.00

     F                                          21.0                                          15.00

The firm should borrow a total of $9.7 million and carry out projects A, B, C, and D. The firm's marginal cost of capital will be 9.35 percent.

3. A firm that pays an interest rate of 9 percent on its bonds has a marginal income tax rate of 50 percent. The interest rate on government bonds is 6 percent, the average rate of return on all stocks traded on the market is 10 percent, the estimated beta coefficient for the firm's stock is 1.5, and the firm intends to raise 60 percent of its capital by borrowing.

(i) What is the firm's cost of debt?

(ii) What is the firm's cost of equity capital?

(iii) What is the firm's composite cost of capital?

Solution:

(i) The cost of debt is the firm's bond rate times one minus the marginal tax rate:

0.09(1 − 0.50) = 0.045

(ii) The cost of equity, by the CAPM, is equal to the risk-free (government bond) rate plus beta times the difference between the risk-free rate and the average return on stocks: 0.06 + (1.5)(0.10 − 0.06) = 0.12

(iii) The composite cost of capital is the weighted average of the cost of debt and the cost of equity: (0.60)(0.045) + (1 − 0.60)(0.12) = 0.075

4. A firm that pays an interest rate of 10 percent on its bonds has a marginal income tax rate of 45 percent. The interest rate on government bonds is 5 percent, the average rate of return on all stocks traded on the market is 11 percent, the estimated beta coefficient for the firm's stock is 2.5, and the firm intends to raise 40 percent of its capital by borrowing.

(i) What is the firm's cost of debt?

(ii) What is the firm's cost of equity capital?

(iii) What is the firm's composite cost of capital?

Solution:

(i) The cost of debt is the firm's bond rate times one minus the marginal tax rate: 0.1(1 − 0.45) = 0.055

(ii) The cost of equity, by the CAPM, is equal to the risk-free (government bond) rate plus beta times the difference between the risk-free rate and the average return on stocks: 0.05 + (2.5)(0.11 − 0.05) = 0.20

(iii) The composite cost of capital is the weighted average of the cost of debt and the cost of equity: (0.40)(0.055) + (1 − 0.40)(0.20) = 0.142

5. A firm is considering two capital investment projects. Project A involves an initial cost of $15,000. The discounted present value of all future cash flows is $18,000. Project B requires an initial expenditure of $25,000. The discounted present value of all future cash flows is $29,000.

(i) Calculate the net present value of each of the two projects. Which would be

preferred according to the net present value criterion?

(ii) Calculate the profitability index of each of the two projects. Which would be

preferred according to the profitability index criterion?

Solution:

(i) The net present value of A is $18,000 − $15,000 = $3,000. The net present value of B is $29,0000 − $25,000 = $4,000. Project B would be preferred to Project A.

(ii) The profitability index of A is $18,000/$15,000 = 1.20. The profitability index of B is $29,000/$25,000 = 1.16. Project A would be preferred to Project B.

6. A firm is considering two capital investment projects. Project A involves an initial cost of $125,000. The discounted present value of all future cash flows is $145,000. Project B requires an initial expenditure of $85,000. The discounted present value of all future cash flows is $102,000.

(i) Calculate the net present value of each of the two projects. Which would be

preferred according to the net present value criterion?

(ii) Calculate the profitability index of each of the two projects. Which would be

preferred according to the profitability index criterion?

Solution:

(i) The net present value of A is $145,000 − $125,000 = $20,000. The net present value of B is $102,000 − $85,000 = $17,000. Project A would be preferred to Project B.

(ii) The profitability index of A is $145,000/$125,000 = 1.16. The profitability index of B is $102,000/$85,000 = 1.20. Project B would be preferred to Project A.

 

7. A firm is considering two alternative projects. Project A requires an initial expenditure of $50,000 plus an expenditure of $10,000 at the end of each of the next five years. It will yield $75,000 in revenue at the end of the first year and at the end of the fifth year. Project B requires an initial expenditure of $100,000. It will yield $40,000 in net revenue at the end of each of the next five years. Both projects have a life of five years with no salvage value or disposal cost. The table below provides present value factors for the firm's discount rate of 12 percent. Calculate the net present value and profitability index

of each project. Which project is preferred by each criterion?

Period               Present value of $1                      Present value of an annuity of $1

     1                            0.8929                                                           0.8929

     2                            0.7972                                                           1.6901

     3                            0.7118                                                           2.4018

     4                            0.6355                                                           3.0373

     5                            0.5674                                                            3.6048

Solution:

NPV of A:

$23,474.5 − 50000 − (10000)(3.6048) + (75000)(0.8929) + (75000)(0.5674)

NPV of B:

$44,192 − 100000 + (3.6048)(40000)

PI of A: $73,474.5/$50,000 = 1.47

PI of B: $144,192/$100,000 = 1.44

The NPV criterion favors B and the PI criterion favors A.

8. A firm is considering two alternative projects. Project A requires an initial expenditure of $100,000 plus an expenditure of $10,000 at the end of each of the next five years. It will yield $98,000 in revenue at the end of the first year and at the end of the fifth year. Project B requires an initial expenditure of $50,000. It will yield $15,000 in net revenue at the end of each of the next five years. Both projects have a life of five years with no salvage value or disposal cost. The table below provides present value factors for the firm's discount rate of 12 percent. Calculate the net present value and profitability index

of each project. Which project is preferred by each criterion?

Period                     Present value of $1                Present value of an annuity of $1

      1                                 0.8929                                                    0.8929

      2                                 0.7972                                                    1.6901

      3                                 0.7118                                                    2.4018

      4                                 0.6355                                                    3.0373

      5                                 0.5674                                                    3.6048

Solution:

NPV of A:

$7061.4 − 100000 − (10000)(3.6048) + (98000)(0.8929) + (98000)(0.5674)

NPV of B:

$4072 − 50000 + (3.6048)(15000)

PI of A: $73,474.5/$50,000 = 1.07

PI of B: $144,192/$100,000 = 1.08

The NPV criterion favors A and the PI criterion favors B.