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13. a. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR for each
project is:
Deepwater Fishing IRR:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
b. To calculate the incremental IRR, we subtract the smaller project’s cash flows from the larger
project’s cash flows. In this case, we subtract the deepwater fishing cash flows from the
submarine ride cash flows. The incremental IRR is the IRR of these incremental cash flows. So,
the incremental cash flows of the submarine ride are:
Year 0
Year 1
Year 2
Year 3
Submarine Ride
–$1,650,000
$1,050,000
$675,000
$520,000
Deepwater Fishing
–835,000
450,000
410,000
335,000
Submarine – Fishing
–$815,000
$600,000
$265,000
$185,000
Setting the present value of these incremental cash flows equal to zero, we find the incremental
IRR is:
evaluating the IRR of each project separately. The IRR decision rule is flawed because there is a
c. The NPV is the sum of the present value of the cash flows from the project, so the NPV of each
project will be:
Deepwater fishing:
NPV = –$835,000 + $450,000/1.15 + $410,000/1.152 + $335,000/1.153
NPV = $86,591.19
14. a. The profitability index is the PV of the future cash flows divided by the initial investment. The
cash flows for both projects are an annuity, so:
PII = $37,000(PVIFA10%,3)/$68,000 = 1.353
b. The NPV of each project is:
NPVI = –$68,000 + $37,000(PVIFA10%,3) = $24,013.52
c. Using the profitability index to compare mutually exclusive projects can be ambiguous when the
magnitudes of the cash flows for the two projects are of different scale. In this problem, Project
15. a. The equation for the NPV of the project is:
NPV = –$67,000,000 + $97,000,000/1.1 – $13,000,000/1.12 = $10,438,016.53
b. The equation for the IRR of the project is:
0 = –$67,000,000 + $97,000,000/(1 + IRR) – $13,000,000/(1 + IRR)2
From Descartes rule of signs, we know there are two possible IRRs since the cash flows change
signs twice. From trial and error, the two IRRs are:
IRR = 29.83%, –85.06%
16. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal
the initial investment.
Board game:
Cumulative cash flows Year 1 = $265,000 = $265,000
Payback period = 1 + ($345,000 – 265,000)/$150,000 = 1.53 years
DVD:
Cumulative cash flows Year 1 = $360,000 = $360,000
Payback period = 1 + ($570,000 – 360,000)/$290,000
Since the board game has a shorter payback period than the DVD project, the company should
choose the board game.
b. The NPV is the sum of the present value of the cash flows from the project, so the NPV of each
project will be:
Board game:
NPV = –$345,000 + $265,000/1.10 + $150,000/1.102 + $98,000/1.103
c. The IRR is the interest rate that makes the NPV of a project equal to zero. So, the IRR of each
project is:
Board game:
0 = –$345,000 + $265,000/(1 + IRR) + $150,000/(1 + IRR)2 + $98,000/(1 + IRR)3
IRR = 24.78%
Since the IRR of the board game is greater than the IRR of the DVD, IRR implies we choose the
board game.
d. To calculate the incremental IRR, we subtract the smaller project’s cash flows from the larger
project’s cash flows. In this case, we subtract the board game cash flows from the DVD cash
flows. The incremental IRR is the IRR of these incremental cash flows. So, the incremental cash
flows of the DVD are:
Year 0
Year 1
Year 2
Year 3
DVD
–$570,000
$360,000
$290,000
$185,000
Board game
–345,000
265,000
150,000
98,000
DVD – Board game
–$225,000
$95,000
$140,000
$87,000
Setting the present value of these incremental cash flows equal to zero, we find the incremental
IRR is:
0 = C0 + C1/(1 + IRR) + C2/(1 + IRR)2 + C3/(1 + IRR)3
0 = –$225,000 + $95,000/(1 + IRR) + $140,000/(1 + IRR)2 + $87,000/(1 + IRR)3
PIG5 = [$23,000,000/1.10 + $29,000,000/1.102 + $21,000,000/1.103]/$35,000,000 = 1.73
PIWi-Fi = [$23,000,000/1.10 + $42,000,000/1.102 + $39,000,000/1.103]/$55,000,000 = 1.54
NPVG5 = –$35,000,000 + $23,000,000/1.10 + $29,000,000/1.102 + $21,000,000/1.103
NPVG5 = $25,653,643.88
NPVWi-Fi = –$55,000,000 + $23,000,000/1.10 + $42,000,000/1.102 + $39,000,000/1.103
NPVWi-Fi = $29,921,111.95
NPVL6 and G5 = $11,141,998.50 + 25,653,643.88
NPVL6 and G5 = $36,795,642.37
This is greater than the Wi-Fi project, so we should accept the L6 project and the G5 project.
Cumulative cash flows Year 1 = $370,000 = $370,000
Cumulative cash flows Year 2 = $370,000 + 310,000 = $680,000
Payback period = 1 + ($650,000 – 370,000)/$310,000 = 1.90 years
Cumulative cash flows Year 2 = $490,000 + 460,000 = $950,000
Payback period = 2 + ($975,000 – 950,000)/$410,000 = 2.06 years
NPVAZM = –$650,000 + $370,000/1.10 + $310,000/1.102 + $260,000/1.103
NPVAZM = $137,903.83
NPVAZF = –$975,000 + $490,000/1.10 + $460,000/1.102 + $410,000/1.103
NPVAZF = $158,658.90
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
And the IRR of the AZF is:
0 = –$975,000 + $490,000/(1 + IRR) + $460,000/(1 + IRR)2 + $410,000/(1 + IRR)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
Year 0
Year 1
Year 2
Year 3
AZF
–$975,000
$490,000
$460,000
$410,000
AZM
–650,000
370,000
310,000
260,000
AZF – AZM
–$325,000
$120,000
$150,000
$150,000
Setting the present value of these incremental cash flows equal to zero, we find the incremental
IRR is:
0 = C0 + C1/(1 + IRR) + C2/(1 + IRR)2 + C3/(1 + IRR)3
0 = –$325,000 + $120,000/(1 + IRR) + $150,000/(1 + IRR)2 + $150,000/(1 + IRR)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
PIB = [$240,000/1.12 + $240,000/1.122]/$350,000 = 1.159
PIC = [$140,000/1.12 + $125,000/1.122]/$200,000 = 1.123
NPVB = –$350,000 + $240,000/1.12 + $240,000/1.122
NPVB = $55,612.24
d. Accept Project B. Since the Projects are mutually exclusive, choose the Project with the highest
PI, while taking into account the scale of the Project. Because Projects A and C have the same
PI of Project A. Because of the problem of scale, we cannot compare the PIs of Projects A and
B. However, we can calculate the PI of the incremental cash flows of the two projects, which are:
Project
C0
C1
C2
B – A
–$150,000
$105,000
$105,000
When calculating incremental cash flows, remember to subtract the cash flows of the project with
the smaller initial cash outflow from those of the project with the larger initial cash outflow. This
procedure insures that the incremental initial cash outflow will be negative. The incremental PI
calculation is:
PI(B – A) = [$105,000/1.12 + $105,000/1.122]/$150,000
PI(B – A) = 1.183
The company should accept Project B since the PI of the incremental cash flows is greater than
e. Remember that the NPV is additive across projects. Since we can spend $550,000, we could take
the initial investment.
Dry Prepeg:
Cumulative cash flows Year 2 = $690,000 + 430,000 = $1,120,000
Cumulative cash flows Year 3 = $690,000 + 430,000 + 1,400,000 = $2,520,000
Payback period = 2 + ($1,800,000 – 690,000 – 430,000)/$1,400,000 = 2.49 years
Solvent Prepeg:
NPVSolvent prepeg = –$925,000 + $565,000/1.10 + $410,000/1.102 + $340,000/1.103
NPVSolvent prepeg = $182,926.37
The NPV criteria implies accepting the dry prepeg because it has the highest NPV.
find that:
IRRDry prepeg = 16.34%
IRRSolvent prepeg = 22.06%
The IRR criteria implies accepting the solvent prepeg because it has the highest IRR. Remember
d. Incremental IRR analysis is necessary. The solvent prepeg has a higher IRR, but is relatively
smaller in terms of investment and NPV. In calculating the incremental cash flows, we subtract
the cash flows from the project with the smaller initial investment from the cash flows of the
Year 0
Year 1
Year 2
Year 3
Dry prepeg
–$1,800,000
$690,000
$430,000
$1,400,000
Solvent prepeg
–925,000
565,000
410,000
340,000
Dry prepeg – Solvent prepeg
–$875,000
$125,000
$20,000
$1,060,000
Setting the present value of these incremental cash flows equal to zero, we find the incremental
IRR is:
0 = –$875,000 + $125,000/(1 + IRR) + $20,000/(1 + IRR)2 + $1,060,000/(1 + IRR)3
find that:
Incremental IRR = 12.33%
For investing-type projects, we accept the larger project when the incremental IRR is greater than
