Financial ACCG253 Management
Tutorial 5 Chapter 8
Net Present and Other Investment Criteria
Solutions to Questions and Problems
Basic
2.
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 $3,400, the payback period is:
Payback = 4 + $360 / $760
Payback = 4.47 years
There is a shortcut to calculate payback period when the future cash flows are an annuity. Just
divide the initial cost by the annual cash flow. For the $3,400 cost, the payback period is:
Payback = $3,400 / $760
Payback = 4.47 years
For an initial cost of $4,450, the payback period is:
Payback = $4,450 / $760
Payback = 5.86 years
The payback period for an initial cost of $6,800 is a little trickier. Notice that the total cash
inflows after eight years will be:
Total cash inflows = 8($760)
Total cash inflows = $6,080
If the initial cost is $6,800, the project never pays back. Notice that if you use the shortcut for
annuity cash flows, you get:
Payback = $6,800 / $760
Payback = 8.95 years
This answer does not make sense since the cash flows stop after ei ght years, so again, we must
conclude the payback period is never
9.
The NPV of a project is the PV of the outflows minus by the PV of the inflows. At a zero
discount rate (and only at a zero discount rate), the cash flows can be added together across ti me.
So, the NPV of the project at a zero percent required return is:
NPV = $36,000 + 14,700 + 19,600 + 13,100
NPV = $11,400
The NPV at a 10 percent required return is:
NPV = $36,000 + $14,700/1.10 + $19,600/1.102 + $13,100/1.103
NPV = $3,404.21
The NPV at a 20 percent required return is:
NPV = $36,000 + $14,700/1.20 + $19,600/1.202 + $13,100/1.203
NPV = $2,557.87
And the NPV at a 30 percent required return is:
NPV = $36,000 + $14,700/1.30 + $19,600/1.302 + $13,100/1.303
NPV = $7,132.00
Notice that as the required return increases, the NPV of the project decreases. This will always
be true for projects with conventional cash flows. Conventional cash flows are negative at the
beginning of the project and positive throughout the rest of the project.
11. The IRR is the interest rate that makes the NPV of the project equal to zero. The equation to
calculate the IRR of Project X is:
0 = $8,000 + $4,300/(1+IRR) + $2,700/(1+IRR)2 + $3,800/(1+IRR)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
IRR = 17.16%
For Project Y, the equation to find the IRR is:
0 = $8,000 + $4,100/(1+IRR) + $2,775/(1+IRR)2 + $3,950/(1+IRR)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
IRR = 16.98%
To find the crossover rate, we subtract the cash flows from one project from the cash flows of the
other project, and find the IRR of the differential cash flows. We will subtract the cash flows
from Project Y from the cash flows from Project X. It is irrelevant which cash flows we subtract
from the other. Subtracting the cash flows, the equation to calculate the IRR for these differential
cash flows is:
Crossover rate: 0 = $200/(1+R) $75/(1+R)2 $150/(1+R)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
R = 7.36%
The table below shows the NPV of each project for r different equired returns. Notice that Project
Y always has a higher NPV for discount rates below 7.36 percent, and always has a lower NPV
for discount rates above 7.36 percent.
R
0%
5%
10%
15%
20%
25%
$NPVX
2,800.00
1,826.80
995.49
279.28
342.59
886.40
$NPVY
2,825.00
1,833.93
988.35
260.71
370.37
921.60
21. At a zero discount rate (and only at a zero discount rate), the cash flows can be added together
across time. So, the NPV of the project at a zero percent required return is:
NPV = $487,160 + 170,605 + 189,895 + 150,387 + 135,867
NPV = $159,594
If the required return is infinite, future cash flows have no value. Even if the cash flow in one
year is $1 trillion, at an infinite rate of interest, the value of this cash flow today is zero. So, if the
future cash flows have no value today, the NPV of the project is simply the cash flow today. So
at an infinite interest rate:
NPV = $487,160
The interest rate that makes the NPV of a project equal to zero is the IRR. The equation for the
IRR of this project is:
0 = $487,160 + 170,605/(1 + IRR) + 189,895/(1 + IRR)2 + 150,387/(1 + IRR)3 + 135,867/(1 +
IRR)4
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
IRR = 13.01%
22. a.
The payback period for each project is:
F:
2 + ($25,000/$80,000) = 2.31 years
G: 3 + ($40,000/$190,000) = 3.21 years
H: 3 + ($10,000/$160,000) = 3.06 years
The payback criterion implies accepting project F because it pays back sooner than Projects G
and H. Projects G and H does not meet the minimum payback of three years.
b.
The NPV for each project is:
F: NPV = $175,000 + $85,000/1.10 + $65,000/1.102 + $80,000/1.103 + $70,000/1.104 +
$55,000/1.105
NPV = $98,058.53
G: NPV = $275,000 + $55,000/1.10 + $70,000/1.102 + $110,000/1.103 + $190,000/1.104 +
$135,000/1.105
NPV = $129,092.80
H: NPV = $200,000 + $60,000/1.10 + $70,000/1.102 + $60,000/1.103 + $160,000/1.104 +
$40,000/1.105
NPV = $91,594.59
NPV criterion implies we should accept project G because project G has a higher NPV than
Projects F and H.
c.
Even though project G does not meet the payback period of three years, it does provide the
largest increase in shareholder wealth, therefore, choose project G. Payback period should
generally be ignored in this situation.
Challenge
32. a.
Here the cash inflows of the project go on forever, which is a perpetuity. Unlike ordinary
perpetuity cash flows, the cash flows here grow at a constant rate forever, which is a
growing perpetuity. If you remember back to the chapter on share valuation, we presented a
formula for valuing a share with constant growth in dividends. This formula is actually the
formula for a growing perpetu ity, so we can use it here. The PV of the future cash flows
from the project is:
PV of cash inflows = C1/(R g)
PV of cash inflows = $60,000/(.13 .06) = $857,142.86
NPV is the PV of the outflows minus by the PV of the inflows, so the NPV is:
NPV of the project = $925,000 + 857,142.86 = $67,857.14
The NPV is negative, so we would reject the project.
b.
Here we want to know the minimum growth rate in cash flows necessary to accept the
project. The minimum growth rate is the growth rate at which we would have a zero NPV.
The equation for a zero NPV, using the equation for the PV of a growing perpetuity is:
0 = $925,000 + $60,000/(.13 g)
Solving for g, we get:
g = .0651 or 6.51%