CHAPTER 5
Why Net Present Value Leads to
Better Investment Decisions Than Other Criteria
Answers to Practice Questions
1.
2.
3.
a.
NPVA   $1000 
$1000
 $90.91
(1.10)
NPVB   $2000 
$1000 $1000 $4000 $1000 $1000




 $4,044.73
(1.10) (1.10) 2 (1.10) 3 (1.10) 4 (1.10)5
NPVC   $3000 
$1000 $1000 $1000 $1000



 $39.47
(1.10) (1.10) 2 (1.10) 4 (1.10) 5
b.
Payback A = 1 year
Payback B = 2 years
Payback C = 4 years
c.
A and B
a.
When using the IRR rule, the firm must still compare the IRR with the
opportunity cost of capital. Thus, even with the IRR method, one must
specify the appropriate discount rate.
b.
Risky cash flows should be discounted at a higher rate than the rate used
to discount less risky cash flows. Using the payback rule is equivalent to
using the NPV rule with a zero discount rate for cash flows before the
payback period and an infinite discount rate for cash flows thereafter.
a.
NPVY   175 
80
70
60
50



 39.06
2
3
1.09 (1.09)
(1.09)
(1.09) 4
NPVZ   350 
125
125
125
125



 54.96
2
3
1.09 (1.09)
(1.09)
(1.09) 4
NPVY   175 
80
70
60
50



 0  IRR Y  19.71%
2
3
1  r (1  r)
(1  r)
(1  r) 4
NPVZ   350 
125
125
125
125



 0  IRR Z  15.97%
2
3
1  r (1  r)
(1  r)
(1  r) 4
NPVZ = €54.96 million, which is positive and greater than NPVY, so project
Z is the better investment.
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b.
NPVY   175 
80
70
60
50



 5.38
2
3
1.18 (1.18)
(1.18)
(1.18) 4
NPVZ   350 
125
125
125
125



  13.74
2
3
1.18 (1.18)
(1.18)
(1.18) 4
NPVZ = – €13.74 million while NPVY = +€5.38 million, so project Y is the
better investment.
c.
The cash flows for (Z – Y) are:
C0 = – €175
C1 =
€45
C2 =
€55
C3 = + €65
C3 = + €75
NPVZ- Y   175 
45
55
65
75



 0  IRR Z- Y  12.73%
2
3
1  r (1  r)
(1  r)
(1  r) 4
The NPV for the larger investment (Project Z) is greater than the NPV for
the smaller investment (Project Y) as long as the cost of capital is less
than 12.73%.
4.
r = -17.44%
0.00%
10.00% 15.00% 20.00% 25.00% 45.27%
Year 0 -3,000.00 -3,000.00 -3,000.00 -3,000.00 -3,000.00 -3,000.00 -3,000.00 -3,000.00
Year 1 3,500.00 4,239.34 3,500.00 3,181.82 3,043.48 2,916.67 2,800.00 2,409.31
Year 2 4,000.00 5,868.41 4,000.00 3,305.79 3,024.57 2,777.78 2,560.00 1,895.43
Year 3 -4,000.00 -7,108.06 -4,000.00 -3,005.26 -2,630.06 -2,314.81 -2,048.00 -1,304.76
PV =
-0.31
500.00
482.35 437.99
379.64 312.00
-0.02
The two IRRs for this project are (approximately): –17.44% and 45.27%
Between these two discount rates, the NPV is positive.
5.
a.
Because Project A requires a larger capital outlay, it is possible that
Project A has both a lower IRR and a higher NPV than Project B. (In fact,
NPVA is greater than NPVB for all discount rates less than 10 percent.)
Because the goal is to maximize shareholder wealth, NPV is the correct
criterion.
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b.
To use the IRR criterion for mutually exclusive projects, calculate the IRR
for the incremental cash flows:
C0
C1
C2
IRR
A-B
-200
+110
+121
10%
Because the IRR for the incremental cash flows exceeds the cost of
capital, the additional investment in A is worthwhile.
c.
6.
a.
NPVA   $400 
$250 $300

 $ 81.86
1.09 (1.09) 2
NPVB   $200 
$140
$179

 $79.10
1.09
(1.09) 2
PID 
PIE 
b.
20,000
1.10  8,182  0 .82
 (  10,000)
10,000
 10,000 
35,000
1.10  11,818  0 .59
 (  20,000)
20,000
 20,000 
Each project has a Profitability Index greater than zero, and so both are
acceptable projects. In order to choose between these projects, we must
use incremental analysis. For the incremental cash flows:
PIE D 
15,000
1.10  3,636  0.36
 (  10,000)
10,000
 10,000 
The increment is thus an acceptable project, and so the larger project
should be accepted, i.e., accept Project E. (Note that, in this case, the
better project has the lower profitability index.)
7.
Use incremental analysis:
Current arrangement
Extra shift
Incremental flows
C1
C2
C3
-2,500,000 -2,500,000 +6,500,000
-5,500,000 +6,500,000
0
-3,000,000 +9,000,000 -6,500,000
The IRRs for the incremental flows are (approximately): 21.13% and 78.87%
If the cost of capital is between these rates, Titanic should work the extra shift.
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8.
Using the fact that Profitability Index = (Net Present Value/Investment), we find:
Project
1
2
3
4
5
6
7
Profitability Index
0.78
-0.07
0.29
0.67
1.06
0.56
0.50
Thus, given the budget of €15million, the best the company can do is to accept
Projects 1, 5, 6 and 7. This precisely uses the entire budget of €15million.
Project 4 has a higher Profitability Index than either project 6 or 7; however, if
Project 4 were accepted, the company would have to reject either Project 6 or
Project 7, which would reduce the total NPV.
If the company accepted all positive NPV projects, the market value (compared
to the market value under the budget limitation) would increase by the NPV of
Project 3 plus the NPV of Project 4: €1.1 million + €1.0 million = €2.1 million.
Thus, the budget limit costs the company €2.1 million in terms of its market
value.
9.
One of the ‘pitfalls’ encountered when using the internal rate of return arises in
the ranking of mutual exclusive projects that have different cash flow patterns
over time. For example, when comparing Projects F and G in Section 5.3 of the
text, we find that the IRR for Project F (33%) is greater than the IRR for Project G
(20%), while the rankings are reversed when comparing net present values. This
inconsistency arises because, at the assumed 10% cost of capital, the later cash
flows from Project G are not discounted as much as they are at higher discount
rates. (This point is clearly demonstrated in Figure 5.5 in the text.) The same
issue prevails when comparing rankings derived from the profitability index with
rankings using the internal rate of return. Since the scale of both projects is the
same (i.e., a $9,000 cash outflow for each project), the profitability index
produces the same rankings as does the NPV criterion, thereby resulting in the
same ‘pitfall’ for the IRR rule.
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Consider the following simple numerical example:
Project
C0
C1
C2
C3
IRR
J
K
- €2
- €2
€3
0
0
0
0
€4
50.00%
25.99%
NPV
(at 10%)
0.727
1.005
PI
(at 10%)
0.364
0.503
If the cost of capital is less than 15.47%, then Project J has the higher NPV and,
therefore, the higher PI. At rates above 15.47%, the more distant cash flow for
Project K is heavily discounted, so that the NPV and the PI for Project J are less
than the NPV and PI, respectively, for Project K. The NPV and PI rankings must
be consistent because the scale of investment is the same for both projects.
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Challenge Questions
1.
The IRR is the discount rate which, when applied to a project’s cash flows, yields
NPV = 0. Thus, it does not represent an opportunity cost. However, if each
project’s cash flows could be invested at that project’s IRR, then the NPV of each
project would be zero because the IRR would then be the opportunity cost of
capital for each project. The discount rate used in an NPV calculation is the
opportunity cost of capital. Therefore, it is true that the NPV rule does assume
that cash flows are reinvested at the opportunity cost of capital.
2.
[Note: In the first printing of the eight edition, Challenge Question 2 refers to
Practice Question 3. The reference should be to Practice Question 4.]
a.
C0 = –3,000
C1 = +3,500
C2 = +4,000
C3 = –4,000
b.
xC 1 
C0 = –3,000
C1 = +3,500
C2 + PV(C3) = +4,000 – 3,571.43 = 428.57
MIRR = 27.84%
C3
xC 2

1.12
1.12 2
(1.122)(xC1) + (1.12)(xC2) = –C3
(x)[(1.122)(C1) + (1.12C2)] = –C3
x 
- C3
(1.12 )(C1 )  (1.12C 2 )
x 
4,000
 0.45
(1.12 )(3,500 )  ( 1.12)(4,00 0)
C0 
2
2
(1 - x)C1
(1 - x)C 2

0
(1  IRR) (1  IRR) 2
 3,000 
(1 - 0.45)(3,50 0) (1 - 0.45)(4,00 0)

0
(1  IRR)
(1  IRR) 2
Now, find MIRR using either trial and error or the IRR function (on a
financial calculator or Excel). We find that MIRR = 23.53%.
It is not clear that either of these modified IRRs is at all meaningful.
Rather, these calculations seem to highlight the fact that MIRR really has
no economic meaning.
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3.
Maximize:
NPV = 6,700xW + 9,000xX + 0XY - 1,500xZ
subject to:
10,000xW + 0xX + 10,000xY + 15,000xZ  20,000
10,000xW + 20,000xX – 5,000xY – 5,000xZ  20,000
0xW – 5,000xX – 5,000xY – 4,000xZ  20,000
0  xW  1
0  xX  1
0  xZ  1
Using Excel Spreadsheet Add-in Linear Programming Module:
Optimized NPV = $13,450
with xW = 1; xX = 0.75; xY = 1 and xZ = 0
If financing available at t = 0 is $21,000:
Optimized NPV = $13,500
with xW = 1; xX = (23/30); xY = 1 and xZ = (2/30)
Here, the shadow price for the constraint at t = 0 is $50, the increase in NPV for a
$1,000 increase in financing available at t = 0.
In this case, the program viewed xZ as a viable choice even though the NPV of
Project Z is negative. The reason for this result is that Project Z provides a
positive cash flow in periods 1 and 2.
If the financing available at t = 1 is $21,000:
Optimized NPV = $13,900
with xW = 1; xX = 0.8; xY = 1 and xZ = 0
Hence, the shadow price of an additional $1,000 in t =1 financing is $450.
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