b.
Since the discounted payback period will always be greater than the undiscounted
payback period when there are positive cash inflows, start the approximation at
year 7.
Cumulative Discounted Cash Flows Year 7
= $150,000 A
7
0.1
=
$730,262.82
Cumulative Discounted Cash Flows Year 8
= $150,000 A
8
0.1
=
$800,238.93
Cumulative Discounted Cash Flows Year 9
= $150,000 A
9
0.1
=
$863,853.57
Cumulative Discounted Cash Flows Year 10
= $150,000 A
10
0.1
= $921,685.07
Cumulative Discounted Cash Flows Year 11
= $150,000 A
11
0.1
= $974,259.15
Cumulative Discounted Cash Flows Year 12
= $150,000 A
12
0.1
= $1,022,053.77
The cumulative discounted cash flows exceed the initial investment of
$1,000,000 by the end of year 12, so the payback period for the project is 12
years.
The discounted payback period is 12 years.
c.
Apply the perpetuity formula, discounted at 10 percent, to calculate the PV of the
annual cash inflows.
NPV
= -$1,000,000 + $150,000 / 0.1
= $500,000
The NPV of the project is $500,000.
7.6
The internal rate of return is the discount rate at which the NPV of the project’s cash
flows equals zero.
Set the project’s cash flows, discounted at the internal rate of return
(IRR), equal to zero.
Solve for the IRR.
IRR(Project A) = C
0
+ C
1
/ (1+IRR) + C
2
/ (1+IRR)
2
0
= -$3,000 + $2,500 / (1+IRR) + $1,000 / (1+IRR)
2
IRR
= 0.1287
IRR(Project B) = C
0
+ C
1
/ (1+IRR) + C
2
/ (1+IRR)
2
0
= -$6,000 + $5,000 / (1+IRR) + $2,000 / (1+IRR)
2
IRR
= 0.1287
Note that since Project B’s cash flows are two times those of Project A, the IRR’s of both
projects are the same.