Solutions to end-of-chapter problems
12-1 Equipment $ 9,000,000
NOWC Investment 3,000,000
Initial investment outlay $12,000,000
12-2 Operating Cash Flow: t = 1
Sales revenues $10,000,000
Operating costs 7,000,000
Depreciation 2,000,000
Operating income before taxes $ 1,000,000
Taxes (40%) 400,000
Operating income after taxes $ 600,000
Add back depreciation 2,000,000
Operating cash flow $ 2,600,000
12-3 Equipment’s original cost $20,000,000
Depreciation (80%) 16,000,000
Book value $ 4,000,000
Gain on sale = $5,000,000 - $4,000,000 = $1,000,000.
Tax on gain = $1,000,000(0.4) = $400,000.
AT net salvage value = $5,000,000 - $400,000 = $4,600,000.
12-4 WACC1 = 12%; WACC2 = 12.5% after $3,250,000 of new capital is raised.
Since each project is independent and of average risk, all projects whose IRR > WACC2 will be accepted. Consequently, Projects A, B, C, D, and E will be accepted and the optimal capital budget is $5,250,000.
12-5 Since Projects C and D are now mutually exclusive only one of them can be accepted. The project with the higher NPV should now be chosen. Therefore, Project D should be selected over Project C. The projects now selected are A, B, D, and E with an optimal capital budget of $4 million.
12-6 Risk-adjusted
Projects Risk WACC IRR Decision
A High 14.5% 14.0% Reject
B Average 12.5 13.5 Accept
C Average 12.5 13.2 Accept
D Average 12.5 13.0 Accept
E Average 12.5 12.7 Accept
F Low 10.5 12.3 Accept
G Low 10.5 12.2 Accept
On the basis of a risk-adjusted WACC, Projects B, C, D, E, F, and G will be accepted and Project A will be rejected. The firm’s optimal capital budget is $6 million.
12-7 Expected NPV = (0.3)(-$10,800) + (0.5)($23,400) + (0.2)($50,400)
= -$3,240 + $11,700 + $10,080
= $18,540.
Since NPV is stated in thousands, E(NPV) = $18,540,000.
NPV = [(0.3)(-$10,800 - $18,540)2 + (0.5)($23,400 - $18,540)2 + (0.2)($50,400 - $18,540)2]½
= [$258,250,680 + $11,809,800 + $203,011,920]½
= $21,750.23.
Since NPV is stated in thousands NPV = $21,750,227.59.
12-8 E(NPV) = 0.05(-$70) + 0.20(-$25) + 0.50($12) + 0.20($20) + 0.05($30)
= -$3.5 + -$5.0 + $6.0 + $4.0 + $1.5
= $3.0 million.
NPV = [0.05(-$70 - $3)2 + 0.20(-$25 - $3)2 + 0.50($12 - $3)2 + 0.20($20 - $3)2 + 0.05($30 - $3)2]½
= $23.622 million.
12-9 a. 0 1 2 3 4 5
Initial investment ($250,000)
Net oper. WC (25,000)
Cost savings $ 90,000 $ 90,000 $ 90,000 $ 90,000 $ 90,000
Depreciation 82,500 112,500 37,500 17,500 0
Oper. inc. before taxes $ 7,500 ($ 22,500) $ 52,500 $ 72,500 $ 90,000
Taxes (40%) 3,000 (9,000) 21,000 29,000 36,000
Oper. Inc. (AT) $ 4,500 ($ 13,500) $ 31,500 $ 43,500 $ 54,000
Add: Depreciation 82,500 112,500 37,500 17,500 0
Oper. CF $ 87,000 $ 99,000 $ 69,000 $ 61,000 $ 54,000
Return of NOWC $ 25,000
Sale of Machine 23,000
Tax on sale (40%) (9,200)
Net cash flow ($275,000) $ 87,000 $ 99,000 $ 69,000 $ 61,000 $ 92,800
NPV = $37,035.13
Notes:
aDepreciation Schedule, Basis = $250,000
MACRS Rate
Basis =
Year Beg. Bk. Value MACRS Rate Depreciation Ending BV
1 $250,000 0.33 $ 82,500 $167,500
2 167,500 0.45 112,500 55,000
3 55,000 0.15 37,500 17,500
4 17,500 0.07 17,500 0
$250,000
b. If savings increase by 20 percent, then savings will be (1.2)($90,000)
= $108,000.
If savings decrease by 20 percent, then savings will be (0.8)($90,000)
= $72,000.
(1) Savings increase by 20%:
0 1 2 3 4 5
Initial investment ($250,000)
Net oper. WC (25,000)
Cost savings $108,000 $108,000 $108,000 $108,000 $108,000
Depreciation 82,500 112,500 37,500 17,500 0
Oper. inc. before taxes $ 25,500 ($ 4,500) $ 70,500 $ 90,500 $108,000
Taxes (40%) 10,200 (1,800) 28,200 36,200 43,200
Oper. Inc. (AT) $ 15,300 ($ 2,700) $ 42,300 $ 54,300 $ 64,800
Add: Depreciation 82,500 112,500 37,500 17,500 0
Oper. CF $ 97,800 $109,800 $ 79,800 $ 71,800 $ 64,800
Return of NOWC $25,000
Sale of Machine 23,000
Tax on sale (40%) (9,200)
Net cash flow ($275,000) $ 97,800 $109,800 $ 79,800 $ 71,800 $103,600
NPV = $77,975.63
(2) Savings decrease by 20%:
0 1 2 3 4 5
Initial investment ($250,000)
Net oper. WC (25,000)
Cost savings $ 72,000 $ 72,000 $ 72,000 $ 72,000 $ 72,000
Depreciation 82,500 112,500 37,500 17,500 0
Oper. inc. before taxes ($ 10,500)($ 40,500) $ 34,500 $ 54,500 $ 72,000
Taxes (40%) (4,200) (16,200) 13,800 21,800 28,800
Oper. Inc. (AT) ($ 6,300)($ 24,300) $ 20,700 $ 32,700 $ 43,200
Add: Depreciation 82,500 112,500 37,500 17,500 0
Oper. CF $ 76,200 $ 88,200 $ 58,200 $ 50,200 $ 43,200
Return of NOWC $25,000
Sale of Machine 23,000
Tax on sale (40%) (9,200)
Net cash flow ($275,000) $ 76,200 $ 88,200 $ 58,200 $ 50,200 $ 82,000
NPV = -$3,905.37
c. Worst-case scenario:
0 1 2 3 4 5
Initial investment ($250,000)
Net oper. WC (30,000)
Cost savings $ 72,000 $ 72,000 $ 72,000 $ 72,000 $ 72,000
Depreciation 82,500 112,500 37,500 17,500 0
Oper. inc. before taxes ($ 10,500)($ 40,500) $ 34,500 $ 54,500 $ 72,000
Taxes (40%) (4,200) (16,200) 13,800 21,800 28,800
Oper. Inc. (AT) ($ 6,300)($ 24,300) $ 20,700 $ 32,700 $ 43,200
Add: Depreciation 82,500 112,500 37,500 17,500 0
Oper. CF $ 76,200 $ 88,200 $ 58,200 $ 50,200 $ 43,200
Return of NOWC $30,000
Sale of Machine 18,000
Tax on sale (40%) (7,200)
Net cash flow ($280,000) $ 76,200 $ 88,200 $ 58,200 $ 50,200 $ 84,000
NPV = -$7,663.52
Base-case scenario:
This was worked out in part a. NPV = $37,035.13.
Best-case scenario:
0 1 2 3 4 5
Initial investment ($250,000)
Net oper. WC ( 20,000)
Cost savings $108,000 $108,000 $108,000 $108,000 $108,000
Depreciation 82,500 112,500 37,500 17,500 0
Oper. inc. before taxes $ 25,500 ($ 4,500) $ 70,500 $ 90,500 $108,000
Taxes (40%) 10,200 (1,800) 28,200 36,200 43,200
Oper. Inc. (AT) $ 15,300 ($ 2,700) $ 42,300 $ 54,300 $ 64,800
Add: Depreciation 82,500 112,500 37,500 17,500 0
Oper. CF $ 97,800 $109,800 $ 79,800 $ 71,800 $ 64,800
Return of NOWC $20,000
Sale of Machine 28,000
Tax on sale (40%) (11,200)
Net cash flow ($270,000) $ 97,800 $109,800 $ 79,800 $ 71,800 $101,600
NPV = $81,733.79
Prob. NPV Prob. NPV
Worst-case 0.35 ($ 7,663.52) ($ 2,682.23)
Base-case 0.35 37,035.13 12,962.30
Best-case 0.30 81,733.79 24,520.14
E(NPV) $34,800.21
NPV = [(0.35)(-$7,663.52 - $34,800.21)2 + (0.35)($37,035.13 - $34,800.21)2 + (0.30)($81,733.79 - $34,800.21)2]½
NPV = [$631,108,927.93 + $1,748,203.59 + $660,828,279.49]½
NPV = $35,967.84.
CV = $35,967.84/$34,800.21 = 1.03.
12-10 a. Time Line (in millions of dollars):
12%
| | | |
-20 3 3 3
NPV = $2.4083 million.
b. Wait 1 year:
NPV @
k = 12%
T
ax
imposed | | | |
|
25% Prob. 0 -20 2.4 2.4 2.4 -$1.8512
Tax not imposed | | | | |
75% Prob. 0 -20 3.2 3.2 3.2 3.4841
Note though, that if the tax is imposed, the NPV of the project is negative and therefore would not be undertaken. The value of this option of waiting one year is evaluated as 0.25($0) + (0.75)($3.4841) = $2.6131 million.
Since the NPV of waiting one year is greater than going ahead and proceeding with the project today, it makes sense to wait.
12-11 a. The net cost is $178,000:
Cost of investment at t = 0:
Base price ($140,000)
Modification (30,000)
Increase in NOWC (8,000)
Cash outlay for new machine ($178,000)
b. The operating cash flows follow:
Year 1 Year 2 Year 3
After-tax savings $30,000 $30,000 $30,000
Depreciation tax savings 22,440 30,600 10,200
Net operating cash flow $52,440 $60,600 $40,200
Notes:
1. The after-tax cost savings is $50,000(1 — T) = $50,000(0.6) = $30,000.
2. The depreciation expense in each year is the depreciable basis, $170,000, times the MACRS allowance percentages of 0.33, 0.45, and 0.15 for Years 1, 2, and 3, respectively. Depreciation expense in Years 1, 2, and 3 is $56,100, $76,500, and $25,500. The depreciation tax savings is calculated as the tax rate (40 percent) times the depreciation expense in each year.
c. The terminal cash flow is $48,760:
Salvage value $60,000
Tax on SV* (19,240)
Return of NOWC 8,000
$48,760
Remaining BV in Year 4 = $170,000(0.07) = $11,900.
*Tax on SV = ($60,000 - $11,900)(0.4) = $19,240.
d. The project has an NPV of ($19,549). Thus, it should not be accepted.
Year Net Cash Flow PV @ 12%
0 ($178,000) ($178,000)
1 52,440 46,821
2 60,600 48,310
3 88,960 63,320
NPV = ($ 19,549)
Alternatively, place the cash flows on a time line:
12%
| | | |
-178,000 52,440 60,600 40,200
48,760
88,960
With a financial calculator, input the appropriate cash flows into the cash flow register, input I = 12, and then solve for NPV = -$19,549.
7-2 a. The net cost is $126,000:
Price ($108,000)
Modification (12,500)
Increase in NOWC (5,500)
Cash outlay for new machine ($126,000)
b. The operating cash flows follow:
Year 1 Year 2 Year 3
After-tax savings $28,600 $28,600 $28,600
Depreciation tax savings 13,918 18,979 6,326
Net operating cash flow $42,518 $47,579 $34,926
Notes:
1. The after-tax cost savings is $44,000(1 - T) = $44,000(0.65)
= $28,600.
2. The depreciation expense in each year is the depreciable basis, $120,500, times the MACRS allowance percentages of 0.33, 0.45, and 0.15 for Years 1, 2, and 3, respectively. Depreciation expense in Years 1, 2, and 3 is $39,765, $54,225, and $18,075. The depreciation tax savings is calculated as the tax rate (35 percent) times the depreciation expense in each year.
c. The terminal cash flow is $50,702:
Salvage value $65,000
Tax on SV* (19,798)
Return of NOWC 5,500
$50,702
BV in Year 4 = $120,500(0.07) = $8,435.
*Tax on SV = ($65,000 - $8,435)(0.35) = $19,798.
d. The project has an NPV of $10,841; thus, it should be accepted.
Year Net Cash Flow PV @ 12%
0 ($126,000) ($126,000)
12%
2 47,579 37,930
3 85,628 60,948
NPV = $ 10,841
Alternatively, place the cash flows on a time line:
0 1 2 3
| | | |
-126,000 42,518 47,579 34,926
50,702
85,628
With a financial calculator, input the appropriate cash flows into the cash flow register, input I = 12, and then solve for NPV = $10,841.
7-5 a. Expected annual cash flows:
Project A: Probable
Probability × Cash Flow = Cash Flow
0.2 $6,000 $1,200
0.6 6,750 4,050
0.2 7,500 1,500
Expected annual cash flow = $6,750
Project B: Probable
Probability × Cash Flow = Cash Flow
0.2 $ 0 $ 0
0.6 6,750 4,050
0.2 18,000 3,600
Expected annual cash flow = $7,650
Coefficient of variation:
Project A:
Project B:
CVA = $474.34/$6,750 = 0.0703.
CVB = $5,797.84/$7,650 = 0.7579.
b. Project B is the riskier project because it has the greater variability in its probable cash flows, whether measured by the standard deviation or the coefficient of variation. Hence, Project B is evaluated at the 12 percent cost of capital, while Project A requires only a 10 percent cost of capital.
NPVA = $6,750(PVIFA10%,3) - $6,750 = $6,750(2.4869) - $6,750
= $16,786.58 - $6,750 = $10,036.58 $10,037.
Alternatively, with a financial calculator, input the appropriate cash flows into the cash flow register, input I = 10, and then solve for NPV = $10,036.25.
NPVB = $7,650(PVIFA12%,3) - $6,750 = $7,650(2.4018) - $6,750
= $18,373.77 - $6,750 = $11,623.77 $11,624.
Alternatively, with a financial calculator, input the appropriate cash flows into the cash flow register, input I = 12, and then solve for NPV = $11,624.01.
Project B has the higher NPV; therefore, the firm should accept Project B.
c. The portfolio effects from Project B would tend to make it less risky than otherwise. This would tend to reinforce the decision to accept Project B. Again, if Project B were negatively correlated with the GDP (Project B is profitable when the economy is down), then it is less risky and Project B's acceptance is reinforced.
7-7 If actual life is 5 years:
Amount Amount Year PV
before after Event Factor
tax tax Occurs at 10% PV
Outflows:
Investment in
new equipment $36,000 $36,000 0 1.0 $36,000
Total PV of outflows $36,000
Inflows:
Operating cash flows
excl. deprec. $12,000 $ 7,200 1-5 3.7908 $27,294
Depreciation 7,200 2,880 1-5 3.7908 10,918
$38,212
NPV = PV(Inflows) - PV(Outflows) = $38,212 - $36,000 = $2,212.
Alternatively, using a time line approach:
10%
| | | | | |
Investment outlay (36,000)
Operating cash flows
excl. deprec. (AT) 7,200 7,200 7,200 7,200 7,200
Depreciation savings 2,880 2,880 2,880 2,880 2,880
Net cash flow (36,000) 10,080 10,080 10,080 10,080 10,080
NPV10% = $2,211.13.
If actual life is 4 years:
Amount Amount Year PV
before after Event Factor
tax tax Occurs at 10% PV
Inflows:
Operating cash flows
excl. deprec. $12,000 $7,200 1-4 3.1699 $22,823
Depreciation 7,200 2,880 1-4 3.1699 9,129
Tax savings on loss
at end of Year 4 7,200 2,880 4 0.6830 1,967
$33,919
NPV = $33,919 - $36,000 = -$2,081.
Alternatively, using a time line approach:
10%
| | | | |
Investment outlay (36,000)
Operating cash flows
excl. deprec. (AT) 7,200 7,200 7,200 7,200
Depreciation savings 2,880 2,880 2,880 2,880
Tax savings on loss 2,880
Net cash flow (36,000) 10,080 10,080 10,080 12,960
NPV10% = -$2,080.68.
If actual life is 8 years:
Amount Amount Year PV
before after Event Factor
tax tax Occurs at 10% PV
Inflows:
Revenues $12,000 $7,200 1-8 5.3349 $38,411
Depreciation 7,200 2,880 1-5 3.7908 10,918
$49,329
NPV = $49,329 - $36,000 = $13,329.
Alternatively, using a time line approach:
10%
| | | | | |
Investment outlay (36,000)
Operating cash flows
excl. deprec. (AT) 7,200 7,200 7,200 7,200 7,200
Depreciation savings 2,880 2,880
Net cash flow (36,000) 10,080 10,080 7,200 7,200 7,200
NPV10% = $13,328.93.
If the life is as low as 4 years (an unlikely event), the investment will not be desirable. But, if the investment life is longer than 4 years, the investment will be a good one. Therefore, the decision will depend on the directors' confidence in the life of the tractor. Given the low proba-bility of the tractor's life being only 4 years, it is likely that the directors will decide to purchase the tractor.
12-15 a. NPV of abandonment after Year t:
Using a financial calculator, input the following: CF0 = -22500, CF1 = 23750, and I = 10 to solve for NPV1 = -$909.09 -$909.
Using a financial calculator, input the following: CF0 = -22500, CF1 = 6250, CF2 = 20250, and I = 10 to solve for NPV2 = -$82.64 -$83.
Using a financial calculator, input the following: CF0 = -22500, CF1 = 6250, Nj = 2, CF3 = 17250, and I = 10 to solve for NPV3 = $1,307.29 $1,307.
Using a financial calculator, input the following: CF0 = -22500, CF1 = 6250, Nj = 3, CF4 = 11250, and I = 10 to solve for NPV4 = $726.73 $727.
Using a financial calculator, input the following: CF0 = -22500, CF1 = 6250, Nj = 5, and I = 10 to solve for NPV5 = $1,192.42 $1,192.
The firm should operate the truck for 3 years, NPV3 = $1,307.
b. No. Abandonment possibilities could only raise NPV and IRR. The firm’s value is maximized by abandoning the project after Year 3.
1
10%
0 1 2 3 4
| | | | |
-8 4 4 4 4
NPV = $4.6795 million.
b. Wait 2 years:
Time Line (in millions of dollars):
NPV @
k = 10%
| | | | | | |
10% Prob. 0 0 -9 2.2 2.2 2.2 2.2 -$1.6746
| | | | | | |
90% Prob. 0 0 -9 4.2 4.2 4.2 4.2 3.5648
If the cash flows are only $2.2 million, the NPV of the project is negative and, thus, would not be undertaken. The value of the option of waiting two years is evaluated as 0.10($0) + 0.90($3.5648) = $3.2083 million.
Since the NPV of waiting two years is less than going ahead and proceeding with the project today, it makes sense to drill today.