Problem 9-2 Calculating Payback [LO2]
|
An investment project provides cash inflows of $660 per year for eight years.
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|
What is the project payback period if the initial cost is $1,525? (Enter 0 if the project never pays back. Round your answer to 2 decimal places, e.g., 32.16.)
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|
Payback period
|
years
|
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What is the project payback period if the initial cost is $3,350? (Enter 0 if the project never pays back. Round your answer to 2 decimal places, e.g., 32.16.)
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Payback period
|
years
|
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What is the project payback period if the initial cost is $5,500? (Enter 0 if the project never pays back. Round your answer to 2 decimal places, e.g., 32.16.)
|
|
Payback period
|
years
|
Explanation:
|
To calculate the payback period, we need to find the time that the project has recovered its initial investment. The cash flows in this problem are an annuity, so the calculation is simpler. If the initial cost is $1,525, the payback period is:
|
|
|
|
Payback = 2 + ($205 / $660) = 2.31 years
|
|
There is a shortcut to calculate the payback period when the future cash flows are an annuity. Just divide the initial cost by the annual cash flow. For the $3,350 cost, the payback period is:
|
|
|
|
Payback = $3,350 / $660 = 5.08 years
|
|
The payback period for an initial cost of $5,500 is a little trickier. Notice that the total cash inflows after eight years will be:
|
|
|
|
Total cash inflows = 8($660) = $5,280
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|
|
|
If the initial cost is $5,500, the project never pays back. Notice that if you use the shortcut for annuity cash flows, you get:
|
|
|
|
Payback = $5,500 / $660 = 8.33 years
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|
|
|
This answer does not make sense since the cash flows stop after eight years, so again, we must conclude the payback period is never.
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Problem 9-7 Calculating IRR [LO5]
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A firm evaluates all of its projects by applying the IRR rule. A project under consideration has the following cash flows:
|
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Year
|
Cash Flow
|
||
|
0
|
–$
|
27,200
|
|
|
1
|
|
11,200
|
|
|
2
|
|
14,200
|
|
|
3
|
|
10,200
|
|
|
|
|||
|
If the required return is 16 percent, what is the IRR for this project? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
|
|
IRR
|
%
|
|
Should the firm accept the project?
|
|
|
|
No
|
Explanation:
|
The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines the IRR for this project is:
|
|
|
|
0 = –$27,200 + $11,200 / (1 + IRR) + $14,200 / (1 + IRR)2 + $10,200 / (1 + IRR)3
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|
|
|
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
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|
|
|
IRR = 14.96%
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|
|
|
Since the IRR is less than the required return, we would reject the project.
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Calculator Solution:
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|
Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.
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|
|
|
|
|
|
CFo
|
–$27,200
|
|
|
C01
|
$11,200
|
|
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F01
|
1
|
|
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C02
|
$14,200
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|
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F02
|
1
|
|
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C03
|
$10,200
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|
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F03
|
1
|
|
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IRR CPT
|
|
|
|
14.96%
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Problem 9-8 Calculating NPV [LO1]
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A firm evaluates all of its projects by applying the NPV decision rule. A project under consideration has the following cash flows:
|
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Year
|
Cash Flow
|
||
|
0
|
–$
|
28,600
|
|
|
1
|
|
12,600
|
|
|
2
|
|
15,600
|
|
|
3
|
|
11,600
|
|
|
|
|||
|
What is the NPV for the project if the required return is 11 percent? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)
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|
NPV
|
$
|
|
At a required return of 11 percent, should the firm accept this project?
|
|
|
|
Yes
|
|
What is the NPV for the project if the required return is 25 percent? (Negative amount should be indicated by a minus sign. Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)
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|
NPV
|
$
|
|
At a required return of 25 percent, should the firm accept this project?
|
|
|
|
No
|
Explanation:
|
The NPV of a project is the PV of the inflows minus the PV of the outflows. The equation for the NPV of this project at an 11 percent required return is:
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|
|
|
NPV = –$28,600 + $12,600 / 1.11 + $15,600 / 1.112 + $11,600 / 1.113 = $3,894.48
|
|
|
|
At an 11 percent required return, the NPV is positive, so we would accept the project.
|
|
|
|
The equation for the NPV of the project at a required return of 25 percent is:
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|
|
|
NPV = –$28,600 + $12,600 / 1.25 + $15,600 / 1.252 + $11,600 / 1.253 = –$2,596.80
|
|
|
|
At a required return of 25 percent, the NPV is negative, so we would reject the project.
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Calculator Solution:
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Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.
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CFo
|
–$28,600
|
CFo
|
–$28,600
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|
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C01
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$12,600
|
C01
|
$12,600
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|
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F01
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1
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F01
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1
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|
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C02
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$15,600
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C02
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$15,600
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|
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F02
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1
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F02
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1
|
|
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C03
|
$11,600
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C03
|
$11,600
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|
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F03
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1
|
F03
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1
|
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I = 11%
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I = 25%
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||
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NPV CPT
|
NPV CPT
|
||
|
|
$3,894.48
|
–$2,596.80
|
||
Problem 9-12 NPV versus IRR [LO1, 5]
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Garage, Inc., has identified the following two mutually exclusive projects:
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Year
|
Cash Flow (A)
|
|
Cash Flow (B)
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||||
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0
|
–$
|
28,500
|
|
|
–$
|
28,500
|
|
|
1
|
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13,900
|
|
|
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4,050
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|
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2
|
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11,800
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9,550
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3
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|
8,950
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|
|
|
14,700
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|
|
4
|
|
4,850
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|
|
|
16,300
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|
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|
|||||||
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a-1
|
What is the IRR for each of these projects? (Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.)
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|
IRR
|
|
Project A
|
%
|
|
Project B
|
%
|
|
|
|
|
a-2
|
Using the IRR decision rule, which project should the company accept?
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|
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Project A
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|
a-3
|
Is this decision necessarily correct?
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|
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No
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b-1
|
If the required return is 11 percent, what is the NPV for each of these projects? (Do not round intermediate calculations and round your answers to 2 decimal places, e.g., 32.16.)
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|
NPV
|
|
Project A
|
$
|
|
Project B
|
$
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|
b-2
|
Which project will the company choose if it applies the NPV decision rule?
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|
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|
Project B
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c.
|
At what discount rate would the company be indifferent between these two projects? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
|
|
Discount rate
|
%
|
Explanation:
a.
|
The IRR is the interest rate that makes the NPV of the project equal to zero. The equation for the IRR of Project A is:
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|
0 = –$28,500 + $13,900 / (1 + IRR) + $11,800 / (1 + IRR)2 + $8,950 / (1 + IRR)3 + $4,850 / (1 + IRR)4
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|
|
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
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|
|
IRR = 17.37%
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|
|
|
The equation for the IRR of Project B is:
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|
|
0 = –$28,500 + $4,050 / (1 + IRR) + $9,550 / (1 + IRR)2 + $14,700 / (1 + IRR)3 + $16,300 / (1 + IRR)4
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|
|
|
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
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|
|
|
IRR = 16.73%
|
|
|
|
Examining the IRRs of the projects, we see that the IRRA is greater than the IRRB, so IRR decision rule implies accepting Project A. This may not be a correct decision; however, because the IRR criterion has a ranking problem for mutually exclusive projects. To see if the IRR decision rule is correct or not, we need to evaluate the project NPVs.
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b.
|
The NPV of Project A is:
|
|
|
|
NPVA = –$28,500 + $13,900 / 1.11 + $11,800 / 1.112 + $8,950 / 1.113 + $4,850 / 1.114
|
|
NPVA = $3,338.68
|
|
|
|
And the NPV of Project B is:
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|
|
|
NPVB = –$28,500 + $4,050 / 1.11 + $9,550 / 1.112 + $14,700 / 1.113 + $16,300 / 1.114
|
|
NPVB = $4,385.47
|
|
|
|
The NPVB is greater than the NPVA, so we should accept Project B.
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c.
|
To find the crossover rate, we subtract the cash flows from one project from the cash flows of the other project. Here, we will subtract the cash flows for Project B from the cash flows of Project A. Once we find these differential cash flows, we find the IRR. The equation for the crossover rate is:
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|
|
Crossover rate: 0 = $9,850 / (1 + R) + $2,250 / (1 + R)2 – $5,750 / (1 + R)3 – $11,450 / (1 + R)4
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|
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
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|
|
R = 15.28%
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|
|
At discount rates above 15.28 percent choose Project A; for discount rates below 15.28 percent choose Project B; indifferent between A and B at a discount rate of 15.28 percent.
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Calculator Solution:
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|
Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.
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|
Project A
|
|
|
|
|
CFo
|
–$28,500
|
CFo
|
–$28,500
|
|
C01
|
$13,900
|
C01
|
$13,900
|
|
F01
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1
|
F01
|
1
|
|
C02
|
$11,800
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C02
|
$11,800
|
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F02
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1
|
F02
|
1
|
|
C03
|
$8,950
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C03
|
$8,950
|
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F03
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1
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F03
|
1
|
|
C04
|
$4,850
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C04
|
$4,850
|
|
F04
|
1
|
F04
|
1
|
|
IRR CPT
|
I = 11%
|
||
|
17.37%
|
NPV CPT
|
||
|
|
$3,338.68
|
||
|
Project B
|
|
|
|
|
CFo
|
–$28,500
|
CFo
|
–$28,500
|
|
C01
|
$4,050
|
C01
|
$4,050
|
|
F01
|
1
|
F01
|
1
|
|
C02
|
$9,550
|
C02
|
$9,550
|
|
F02
|
1
|
F02
|
1
|
|
C03
|
$14,700
|
C03
|
$14,700
|
|
F03
|
1
|
F03
|
1
|
|
C04
|
$16,300
|
C04
|
$16,300
|
|
F04
|
1
|
F04
|
1
|
|
IRR CPT
|
I = 11%
|
||
|
16.73%
|
NPV CPT
|
||
|
|
$4,385.47
|
||
|
Crossover rate
|
|
|
CFo
|
$0
|
|
C01
|
$9,850
|
|
F01
|
1
|
|
C02
|
$2,250
|
|
F02
|
1
|
|
C03
|
–$5,750
|
|
F03
|
1
|
|
CO4
|
–$11,450
|
|
FO4
|
1
|
|
IRR CPT
|
|
|
15.28%
|
|
Problem 10-9 Calculating Project OCF [LO1]
|
Quad Enterprises is considering a new three-year expansion project that requires an initial fixed asset investment of $2.67 million. The fixed asset will be depreciated straight-line to zero over its three-year tax life, after which time it will be worthless. The project is estimated to generate $2,070,000 in annual sales, with costs of $765,000. If the tax rate is 34 percent, what is the OCF for this project? (Do not round intermediate calculations. Enter your answer in dollars, not millions of dollars, e.g. 1,234,567.)
|
|
OCF
|
$
|
Explanation:
|
Using the tax shield approach to calculating OCF (Remember the approach is irrelevant; the final answer will be the same no matter which of the four methods you use.), we get:
|
|
OCF = (Sales − Costs)(1 − TC) + TC(Depreciation)
|
|
OCF = ($2,070,000 − 765,000)(1 − .34) + .34($2,670,000 / 3)
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|
OCF = $1,163,900
|
Problem 10-10 Calculating Project NPV [LO1]
|
Quad Enterprises is considering a new three-year expansion project that requires an initial fixed asset investment of $2.79 million. The fixed asset will be depreciated straight-line to zero over its three-year tax life, after which time it will be worthless. The project is estimated to generate $2,110,000 in annual sales, with costs of $805,000. The tax rate is 35 percent and the required return on the project is 12 percent. What is the project’s NPV? (Enter your answer in dollars, not millions of dollars, e.g. 1,234,567. Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)
|
|
NPV
|
$
|
Explanation:
|
Using the tax shield approach to calculating OCF (Remember the approach is irrelevant; the final answer will be the same no matter which of the four methods you use.), we get:
|
|
OCF = (Sales − Costs)(1 − TC) + TC(Depreciation)
|
|
OCF = ($2,110,000 − 805,000)(1 − .35) + .35($2,790,000 / 3)
|
|
OCF = $1,173,750
|
|
Since we have the OCF, we can find the NPV as the initial cash outlay plus the PV of the OCFs, which are an annuity, so the NPV is:
|
|
NPV = −$2,790,000 + $1,173,750(PVIFA12%,3)
|
|
NPV = $29,149.45
|
Problem 10-11 Calculating Project Cash Flow from Assets [LO1]
|
Quad Enterprises is considering a new three-year expansion project that requires an initial fixed asset investment of $2.4 million. The fixed asset will be depreciated straight-line to zero over its three-year tax life. The project is estimated to generate $1,980,000 in annual sales, with costs of $675,000. The project requires an initial investment in net working capital of $200,000, and the fixed asset will have a market value of $310,000 at the end of the project. If the tax rate is 34 percent, what is the project’s Year 0 net cash flow? Year 1? Year 2? Year 3? (Do not round intermediate calculations. Enter your answers in dollars, not millions of dollars, e.g. 1,234,567. Negative amounts should be indicated by a minus sign.)
|
|
Years
|
Cash Flow
|
|
Year 0
|
$
|
|
Year 1
|
$
|
|
Year 2
|
$
|
|
Year 3
|
$
|
|
|
|
|
If the required return is 18 percent, what is the project's NPV? (Do not round intermediate calculations and round your final answer to 2 decimal places, e.g., 32.16.)
|
|
NPV
|
$
|
rev: 11_04_2015_QC_CS-32334
Explanation:
|
The Year 0 cash flow is:
|
|
|
|
Year 0 = −$2,400,000 − 200,000 = –$2,600,000
|
|
|
|
The cash outflow at the beginning of the project will increase because of the spending on NWC. At the end of the project, the company will recover the NWC, so it will be a cash inflow. The sale of the equipment will result in a cash inflow, but we also must account for the taxes that will be paid on this sale. So, the cash flows for each year of the project will be:
|
|
OCF = (Sales − Costs)(1 − TC) + TC(Depreciation)
|
|
OCF = ($1,980,000 − 675,000)(1 − .34) + .34($2,400,000 / 3)
|
|
OCF = $1,133,300
|
|
In Years 1 and 2, the only cash flow is the OCF. In Year 3, the total cash flow will include the recovery of the NWC and the aftertax salvage value, so:
|
|
|
|
Year 3 = $1,133,300 + 200,000 + 310,000 + ($0 − 310,000)(.34)
|
|
Year 3 = $1,537,900
|
|
And the NPV of the project is:
|
|
NPV = −$2,600,000 + $1,133,300(PVIFA18%,2) + ($1,537,900 / 1.183)
|
|
NPV = $110,355.56
|
Problem 12-7 Calculating Returns and Variability [LO1]
|
|
Returns
|
|
|||||
|
Year
|
X
|
Y
|
|
||||
|
1
|
|
15
|
%
|
|
22
|
%
|
|
|
2
|
|
29
|
|
|
30
|
|
|
|
3
|
|
10
|
|
|
10
|
|
|
|
4
|
–
|
22
|
|
–
|
27
|
|
|
|
5
|
|
10
|
|
|
21
|
|
|
|
|
|||||||
|
Using the returns shown above, calculate the arithmetic average returns, the variances, and the standard deviations for X and Y. (Do not round intermediate calculations. Enter your average return and standard deviation as a percent rounded to 2 decimal places, e.g., 32.16, and round the variance to 5 decimal places, e.g., 32.16161.)
|
|
|
X
|
Y
|
|
Average return
|
%
|
%
|
|
Variance
|
|
|
|
Standard deviation
|
%
|
%
|
|
|
||
rev: 09_29_2015_QC_CS-27339
Problem 12-8 Risk Premiums [LO2, 3]
|
Suppose we have the following returns for large-company stocks and Treasury bills over a six year period:
|
|
Year
|
Large Company
|
US Treasury Bill
|
|
1
|
3.95
|
6.53
|
|
2
|
14.13
|
4.38
|
|
3
|
19.07
|
4.25
|
|
4
|
–14.61
|
7.30
|
|
5
|
–32.10
|
4.94
|
|
6
|
37.32
|
6.14
|
|
|
||
|
a.
|
Calculate the arithmetic average returns for large-company stocks and T-bills over this period. (Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.)
|
|
|
Average returns
|
|
Large company stocks
|
%
|
|
T-bills
|
%
|
|
|
|
|
b.
|
Calculate the standard deviation of the returns for large-company stocks and T-bills over this period. (Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.)
|
|
|
Standard deviation
|
|
Large company stocks
|
%
|
|
T-bills
|
%
|
|
|
|
|
c-1
|
Calculate the observed risk premium in each year for the large-company stocks versus the T-bills. What was the average risk premium over this period? (Negative amount should be indicated by a minus sign. Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
|
|
Average risk premium
|
%
|
|
c-2
|
Calculate the observed risk premium in each year for the large-company stocks versus the T-bills. What was the standard deviation of the risk premium over this period? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
|
|
Standard deviation
|
%
|
Explanation:
|
We will calculate the sum of the returns for each asset and the observed risk premium first. Doing so, we get:
|
|
Year
|
|
Large Co. Stock Return
|
|
T-Bill Return
|
|
Risk Premium
|
|
||||||
|
1
|
|
|
3.95
|
%
|
|
|
6.53
|
%
|
|
–
|
2.58
|
%
|
|
|
2
|
|
|
14.13
|
|
|
|
4.38
|
|
|
|
9.75
|
|
|
|
3
|
|
|
19.07
|
|
|
|
4.25
|
|
|
|
14.82
|
|
|
|
4
|
|
–
|
14.61
|
|
|
|
7.30
|
|
|
–
|
21.91
|
|
|
|
5
|
|
–
|
32.10
|
|
|
|
4.94
|
|
|
–
|
37.04
|
|
|
|
6
|
|
|
37.32
|
|
|
|
6.14
|
|
|
|
31.18
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
27.76
|
|
|
|
33.54
|
|
|
–
|
5.78
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||
a.
|
The average return for large company stocks over this period was:
|
|
|
|
Large company stocks average return = 27.76% / 6 = 4.63%
|
|
|
|
And the average return for T-bills over this period was:
|
|
|
|
T-bills average return = 33.54% / 6 = 5.59%
|
b.
|
Using the equation for variance, we find the variance for large company stocks over this period was:
|
|
|
|
Variance = 1/5[(.0395 – .0463)2 + (.1413 – .0463)2 + (.1907 – .0463)2 + (– .1461 – .0463)2 + (–.3210 – .0463)2 + (.3732 – .0463)2]
|
|
|
|
Variance = .061743
|
|
|
|
And the standard deviation for large company stocks over this period was:
|
|
|
|
Standard deviation = (.061743)1/2 = .2485, or 24.85%
|
|
|
|
Using the equation for variance, we find the variance for T-bills over this period was:
|
|
|
|
Variance = 1/5[(.0653 – .0559)2 + (.0438 – .0559)2 + (.0425 – .0559)2 + (.0730 – .0559)2 + (.0494 – .0559)2 + (.0614 – .0559)2]
|
|
|
|
Variance = .000156
|
|
|
|
And the standard deviation for T-bills over this period was:
|
|
|
|
Standard deviation = (.000156)1/2 = .0125, or 1.25%
|
c.
|
The average observed risk premium over this period was:
|
|
|
|
Average observed risk premium = –5.78% / 6 = –.96%
|
|
|
|
The variance of the observed risk premium was:
|
|
|
|
Variance = 1/5[(–.0258 – (–.0096))2 + (.0975 – (–.0096))2 + (.1482 – (–.0096))2 + (–.2191 – (–.0096))2 + (–.3704 – (–.0096))2 + (.3118 – (–.0096))2]
|
|
|
|
Variance = .062800
|
|
|
|
And the standard deviation of the observed risk premium was:
|
|
|
|
Standard deviation = (.062800)1/2 = .2506, or 25.06%
|
Problem 12-9 Calculating Returns and Variability [LO1]
|
You’ve observed the following returns on Crash-n-Burn Computer’s stock over the past five years: 14 percent, –7 percent, 17 percent, 15 percent, and 10 percent.
|
|
a.
|
What was the arithmetic average return on Crash-n-Burn’s stock over this five-year period? (Do not round intermediate calculations. Enter your answer as a percent rounded to 1 decimal place, e.g., 32.1.)
|
|
Average return
|
%
|
|
b-1
|
What was the variance of Crash-n-Burn’s returns over this period? (Do not round intermediate calculations and round your answer to 5 decimal places, e.g., 32.16161.)
|
|
Variance
|
|
|
b-2
|
What was the standard deviation of Crash-n-Burn’s returns over this period? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
|
|
Standard deviation
|
%
|
Explanation:
a.
|
To find the average return, we sum all the returns and divide by the number of returns, so:
|
|
|
|
Average return = (.14 – .07 + .17 + .15 + .10) / 5
|
|
Average return = .098, or 9.8%
|
b.
|
Using the equation to calculate variance, we find:
|
|
|
|
Variance = 1/4[(.14 – .098)2 + (–.07 – .098)2 + (.17 – .098)2 + (.15 – .098)2 + (.10 – .098)2]
|
|
Variance = .00947
|
|
So, the standard deviation is:
|
|
|
|
Standard deviation = (.00947)1/2
|
|
Standard deviation = .0973, or 9.73%
|
Problem 12-11 Calculating Real Rates [LO1]
|
You’ve observed the following returns on Crash-n-Burn Computer’s stock over the past five years: 18 percent, –14 percent, 20 percent, 22 percent, and 10 percent. Suppose the average inflation rate over this period was 3.1 percent and the average T-bill rate over the period was 4.4 percent.
|
|
What was the average real risk-free rate over this time period? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
|
|
Average real risk-free rate
|
%
|
|
What was the average real risk premium? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
|
|
Average real risk premium
|
%
|
Explanation:
|
We can find the average real risk-free rate using the Fisher equation. The average real risk-free rate was:
|
|
|
|
(1 + R) = (1 + r)(1 + h)
|
|
|
|
f = (1.044 / 1.031) – 1
|
|
f = .0126, or 1.26%
|
|
|
|
To find the average return, we sum all the returns and divide by the number of returns, so:
|
|
|
|
Average return = (.18 – .14 + .20 + .22 + .10) / 5
|
|
Average return = .112, or 11.2%
|
|
|
|
To calculate the average real return, we can use the average return of the asset, and the average inflation in the Fisher equation. Doing so, we find:
|
|
|
|
(1 + R) = (1 + r)(1 + h)
|
|
|
|
= (1.112 / 1.031) – 1
|
|
= .0786, or 7.86%
|
|
|
|
And to calculate the average real risk premium, we can subtract the average risk-free rate from the average real return. So, the average real risk premium was:
|
|
|
|
Average real risk premium = Average real return – Average risk-free rate
|
|
Average real risk premium = 7.86% – 1.26%
|
|
Average real risk premium = 6.60%
|
Problem 12-15 Arithmetic and Geometric Returns [LO1]
|
A stock has had returns of 6 percent, 24 percent, 16 percent, −12 percent, 31 percent, and −6 percent over the last six years.
|
|
What are the arithmetic and geometric returns for the stock? (Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.)
|
|
|
|
|
Arithmetic return
|
%
|
|
Geometric return
|
%
|
|
|
|
Explanation:
|
The arithmetic average return is the sum of the known returns divided by the number of returns, so:
|
|
Arithmetic average return = (.06 + .24 + .16 – .12 + .31 – .06) / 6
|
|
Arithmetic average return = .0983, or 9.83%
|
|
Using the equation for the geometric return, we find:
|
|
Geometric average return = [(1 + R1) × (1 + R2) × … × (1 + RT)]1/T – 1
|
|
Geometric average return = [(1 + .06)(1 + .24)(1 + .16)(1 – .12)(1 + .31)(1 – .06)](1/6) – 1
|
|
Geometric average return = .0873, or 8.73%
|
|
Remember, the geometric average return will always be less than the arithmetic average return if the returns have any variation.
|
Problem 13-7 Calculating Returns and Standard Deviations [LO1]
|
Consider the following information:
|
|
|
|
Rate of Return If State Occurs
|
|||||||
|
State of
|
Probability of
|
|
|||||||
|
Economy
|
State of Economy
|
Stock A
|
Stock B
|
||||||
|
Recession
|
|
.15
|
|
|
.06
|
|
−
|
.19
|
|
|
Normal
|
|
.60
|
|
|
.09
|
|
|
.10
|
|
|
Boom
|
|
.25
|
|
|
.14
|
|
|
.27
|
|
|
|
|||||||||
|
Calculate the expected return for each stock. (Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.)
|
|
|
Expected return
|
|
Stock A
|
%
|
|
Stock B
|
%
|
|
|
|
|
Calculate the standard deviation for each stock. (Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.)
|
|
|
Standard deviation
|
|
Stock A
|
%
|
|
Stock B
|
%
|
|
|
|
Explanation:
|
The expected return of an asset is the sum of each return times the probability of that return occurring. So, the expected return of each stock asset is:
|
|
E(RA) = .15(.06) + .60(.09) + .25(.14)
|
|
E(RA) = .0980, or 9.80%
|
|
E(RB) = .15(−.19) + .60(.10) + .25(.27)
|
|
E(RB) = .0990, or 9.90%
|
|
To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, then add all of these up. The result is the variance. So, the variance and standard deviation of each stock is:
|
|
σA2 =.15(.06 − .0980)2 + .60(.09 − .0980)2 + .25(.14 − .0980)2
|
|
σA2 = .00070
|
|
σA = (.00070)1/2
|
|
σA = .0264, or 2.64%
|
|
σB2 =.15(−.19 − .0990)2 + .60(.10 − .0990)2 + .25(.27 − .0990)2
|
|
σB2 = .01984
|
|
σB = (.01984)1/2
|
|
σB = .1409, or 14.09%
|
Problem 13-12 Calculating Portfolio Betas [LO4]
|
You own a portfolio equally invested in a risk-free asset and two stocks. If one of the stocks has a beta of 1.11 and the total portfolio is equally as risky as the market, what must the beta be for the other stock in your portfolio? (Do not round intermediate calculations. Round your answer to 2 decimal places, e.g., 32.16.)
|
|
Portfolio beta
|
|
Explanation:
|
The beta of a portfolio is the sum of the weight of each asset times the beta of each asset. If the portfolio is as risky as the market, it must have the same beta as the market. Since the beta of the market is one, we know the beta of our portfolio is one. We also need to remember that the beta of the risk-free asset is zero. It has to be zero since the asset has no risk. Setting up the equation for the beta of our portfolio, we get:
|
|
βP = 1.0 = 1/3(0) + 1/3(1.11) + 1/3(βX)
|
|
Solving for the beta of Stock X, we get:
|
|
βX = 1.89
|
Problem 13-17 Using the SML [LO4]
|
Asset W has an expected return of 12.2 percent and a beta of 2.00. If the risk-free rate is 4.1 percent, complete the following table for portfolios of Asset W and a risk-free asset. (Leave no cells blank - be certain to enter "0" wherever required. Do not round intermediate calculations. Enter your expected returns as a percent rounded to 2 decimal places, e.g., 32.16, and your beta answers to 3 decimal places, e.g., 32.161.)
|
|
Percentage of Portfolio
in Asset W |
Portfolio
Expected Return |
Portfolio
Beta |
|||
|
0
|
%
|
%
|
|
|
|
|
25
|
|
%
|
|
|
|
|
50
|
|
%
|
|
|
|
|
75
|
|
%
|
|
|
|
|
100
|
|
%
|
|
|
|
|
125
|
|
%
|
|
|
|
|
150
|
|
%
|
|
|
|
|
|
|||||
|
If you plot the relationship between portfolio expected return and portfolio beta, what is the slope of the line that results? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
|
|
Slope of the line
|
%
|
Explanation:
|
First, we need to find the β of the portfolio. The β of the risk-free asset is zero, and the weight of the risk-free asset is one minus the weight of the stock, so the β of the portfolio is:
|
|
βP = wW(2.00) + (1 − wW)(0) = 2.00wW
|
|
So, to find the β of the portfolio for any weight of the stock, we simply multiply the weight of the stock times its β.
|
|
Even though we are solving for the β and expected return of a portfolio of one stock and the risk-free asset for different portfolio weights, we are really solving for the SML. Any combination of this stock and the risk-free asset will fall on the SML. For that matter, a portfolio of any stock and the risk-free asset, or any portfolio of stocks, will fall on the SML. We know the slope of the SML line is the market risk premium, so using the CAPM and the information concerning this stock, the market risk premium is:
|
|
E(RW) = .122 = .041 + MRP(2.00)
|
|
MRP = .081 / 2.00
|
|
MRP = .0405, or 4.05%
|
|
So, now we know the CAPM equation for any stock is:
|
|
E(RP) = .041 + .0405βp
|
|
The slope of the SML is equal to the market risk premium, which is .0405, or 4.05%
|
Problem 13-19 Reward-to-Risk Ratios [LO4]
|
Stock Y has a beta of .9 and an expected return of 11.2 percent. Stock Z has a beta of .5 and an expected return of 7.2 percent.
|
|
What would the risk-free rate have to be for the two stocks to be correctly priced? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
|
|
Risk-free rate
|
%
|
Explanation:
|
We need to set the reward-to-risk ratios of the two assets equal to each other, which is:
|
|
|
|
(.112 – Rf) / .9 = (.072 – Rf) / .5
|
|
|
|
We can cross multiply to get:
|
|
|
|
.5(.112 – Rf) = .9(.072 – Rf)
|
|
|
|
Solving for the risk-free rate, we find:
|
|
|
|
.0560 – .5Rf = .0648 – .9Rf
|
|
|
|
Rf = .0220, or 2.20%
|
Problem 14-13 Calculating the WACC [LO3]
|
Dinklage Corp. has 6 million shares of common stock outstanding. The current share price is $85, and the book value per share is $8. The company also has two bond issues outstanding. The first bond issue has a face value of $65 million, a coupon rate of 8 percent, and sells for 95 percent of par. The second issue has a face value of $40 million, a coupon rate of 9 percent, and sells for 108 percent of par. The first issue matures in 23 years, the second in 5 years.
|
|
|
|
Suppose the most recent dividend was $5.70 and the dividend growth rate is 4 percent. Assume that the overall cost of debt is the weighted average of that implied by the two outstanding debt issues. Both bonds make semiannual payments. The tax rate is 38 percent. What is the company’s WACC? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
|
|
WACC
|
%
|
Explanation:
|
The market value of equity is the share price times the number of shares, so:
|
|
|
|
MVE = 6,000,000($85)
|
|
MVE = $510,000,000
|
|
|
|
Using the relationship that the total market value of debt is the price quote times the par value of the bond, we find the market value of debt is:
|
|
|
|
MVD = .95($65,000,000) + 1.08($40,000,000)
|
|
MVD = $104,950,000
|
|
|
|
This makes the total market value of the company:
|
|
|
|
V = $510,000,000 + 104,950,000
|
|
V = $614,950,000
|
|
|
|
And the market value weights of equity and debt are:
|
|
|
|
E/V = $510,000,000 / $614,950,000
|
|
E/V = .8293
|
|
|
|
D/V = 1 – E/V = .1707
|
|
Next, we will find the cost of equity for the company. The information provided allows us to solve for the cost of equity using the dividend growth model, so:
|
|
|
|
RE = [$5.70(1.04) / $85] + .04
|
|
RE = .1097, or 10.97%
|
|
|
|
Next, we need to find the YTM on both bond issues. Doing so, we find:
|
|
|
|
P1 = $950 = $40(PVIFAR%,46) + $1,000(PVIFR%,46)
|
|
R = 4.249%
|
|
YTM = 4.249% × 2 = 8.50%
|
|
|
|
P2 = $1,080 = $45(PVIFAR%,10) + $1,000(PVIFR%,10)
|
|
R = 3.536%
|
|
YTM = 3.536% × 2 = 7.07%
|
|
|
|
To find the weighted average aftertax cost of debt, we need the weight of each bond as a percentage of the total debt. We find:
|
|
|
|
xD1 = .95($65,000,000) / $104,950,000
|
|
xD1 = .5884
|
|
|
|
xD2 = 1.08($40,000,000) / $104,950,000
|
|
xD2 = .4116
|
|
|
|
Now we can multiply the weighted average cost of debt times one minus the tax rate to find the weighted average aftertax cost of debt. This gives us:
|
|
|
|
RD = (1 – .38)[(.5884)(.0850) + (.4116)(.0707)]
|
|
RD = .0491, or 4.91%
|
|
|
|
Using these costs and the weight of debt we calculated earlier, the WACC is:
|
|
|
|
WACC = .8293(.1097) + .1707(.0491)
|
|
WACC = .0994, or 9.94%
|
Problem 14-15 Finding the WACC [LO3]
|
You are given the following information for Watson Power Co. Assume the company’s tax rate is 30 percent.
|
|
Debt:
|
9,000 6.4 percent coupon bonds outstanding, $1,000 par value, 20 years to maturity, selling for 107 percent of par; the bonds make semiannual payments.
|
|
|
|
|
Common stock:
|
360,000 shares outstanding, selling for $54 per share; the beta is 1.10.
|
|
|
|
|
Preferred stock:
|
14,000 shares of 4 percent preferred stock outstanding, currently selling for $74 per share.
|
|
|
|
|
Market:
|
11 percent market risk premium and 4.4 percent risk-free rate.
|
|
What is the company's WACC? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
|
|
WACC
|
%
|
Explanation:
|
We will begin by finding the market value of each type of financing. We find:
|
|
|
|
MVD = 9,000($1,000)(1.07) = $9,630,000
|
|
MVE = 360,000($54) = $19,440,000
|
|
MVP = 14,000($74) = $1,036,000
|
|
And the total market value of the firm is:
|
|
|
|
V = $9,630,000 + 19,440,000 + 1,036,000
|
|
V = $30,106,000
|
|
|
|
Now, we can find the cost of equity using the CAPM. The cost of equity is:
|
|
|
|
RE = .044 + 1.10(.11)
|
|
RE = .1650, or 16.50%
|
|
|
|
The cost of debt is the YTM of the bonds, so:
|
|
|
|
P0 = $1,070 = $32.00(PVIFAR%,40) + $1,000(PVIFR%,40)
|
|
R = 2.902%
|
|
YTM = 2.902% × 2 = 5.80%
|
|
|
|
And the aftertax cost of debt is:
|
|
|
|
RD = (1 – .30)(.0580)
|
|
RD = .0406, or 4.06%
|
|
|
|
The cost of preferred stock is:
|
|
|
|
RP = $4 / $74
|
|
RP = .0541, or 5.41%
|
|
|
|
Now we have all of the components to calculate the WACC. The WACC is:
|
|
|
|
WACC = .0406(9.630 / 30.106) + .1650(19.440 / 30.106) + .0541(1.036 / 30.106)
|
|
WACC = .1214, or 12.14%
|
|
|
|
Notice that we didn’t include the (1 – TC) term in the WACC equation. We used the aftertax cost of debt in the equation, so the term is not needed here.
|
Problem 14-16 Finding the WACC [LO3]
|
Titan Mining Corporation has 9.7 million shares of common stock outstanding, 410,000 shares of 4 percent preferred stock outstanding, and 215,000 8.5 percent semiannual bonds outstanding, par value $1,000 each. The common stock currently sells for $45 per share and has a beta of 1.35, the preferred stock currently sells for $95 per share, and the bonds have 10 years to maturity and sell for 116 percent of par. The market risk premium is 8.5 percent, T-bills are yielding 5 percent, and the company’s tax rate is 35 percent.
|
|
a.
|
What is the firm’s market value capital structure? (Do not round intermediate calculations. Round your answers to 4 decimal places, e.g., 32.1616.)
|
|
|
Market value weight
|
|
Debt
|
|
|
Preferred stock
|
|
|
Equity
|
|
|
|
|
|
b.
|
If the company is evaluating a new investment project that has the same risk as the firm’s typical project, what rate should the firm use to discount the project’s cash flows? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
|
|
Discount rate
|
%
|
Explanation:
a.
|
We will begin by finding the market value of each type of financing. We find:
|
|
|
|
MVD = 215,000($1,000)(1.16) = $249,400,000
|
|
MVE = 9,700,000($45) = $436,500,000
|
|
MVP = 410,000($95) = $38,950,000
|
|
And the total market value of the firm is:
|
|
|
|
V = $249,400,000 + 436,500,000 + 38,950,000 = $724,850,000
|
|
|
|
So, the market value weights of the company’s financing are:
|
|
|
|
D/V = $249,400,000 / $724,850,000 = .3441
|
|
P/V = $38,950,000 / $724,850,000 = .0537
|
|
E/V = $436,500,000 / $724,850,000 = .6022
|
b.
|
For projects equally as risky as the firm itself, the WACC should be used as the discount rate.
|
|
|
|
First we can find the cost of equity using the CAPM. The cost of equity is:
|
|
|
|
RE = .05 + 1.35(.085)
|
|
RE = .1648, or 16.48%
|
|
|
|
The cost of debt is the YTM of the bonds, so:
|
|
|
|
P0 = $1,160 = $42.50(PVIFAR%,20) + $1,000(PVIFR%,20)
|
|
R = 3.159%
|
|
YTM = 3.159% × 2 = 6.32%
|
|
|
|
And the aftertax cost of debt is:
|
|
|
|
RD = (1 – .35)(.0632)
|
|
RD = .0411, or 4.11%
|
|
|
|
The cost of preferred stock is:
|
|
|
|
RP = $4 / $95
|
|
RP = .0421, or 4.21%
|
|
|
|
Now we can calculate the WACC as:
|
|
|
|
WACC = .3441(.0411) + .0537(.0421) + .6022(.1648)
|
|
WACC = .1156, or 11.56%
|
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