CHAPTER 6
ANSWERS
6-1
Project classification schemes can be used to indicate how much analysis is required to evaluate a
given project, the level of the executive who must approve the project, and the required rate of
return that should be used to calculate the project’s NPV. Thus, classification schemes can increase
the efficiency of the capital budgeting process.
6-2
The NPV is obtained by discounting future cash flows, and the discounting process actually
compounds the interest rate over time. Thus, a change in the discount rate has a much greater
impact on a cash flow in Year 5 than on a cash flow in Year 1.
6-3
This question is related to Question 6-2 and the same rationale applies. With regard to the second
part of the question, the answer is no; the IRR rankings are constant and independent of the firm’s
required rate of return.
6-4
The NPV and IRR methods both involve compound interest, and the mathematics of discounting
requires an assumption about reinvestment rates. The NPV method assumes reinvestment at the
required rate of return, whereas the IRR method assumes reinvestment at the IRR.
6-5
The statement is true. The NPV and IRR methods result in conflicts only if mutually exclusive
projects are being considered because the NPV is positive if and only if the IRR is greater than the
required rate of return. If the assumptions were changed so that the firm had mutually exclusive
projects, then the IRR and NPV methods could lead to different conclusions. A change in the
required rate of return or in the cash flow streams would not lead to conflicts if the projects were
independent. Therefore, the IRR method can be used in lieu of the NPV if the projects being
considered are independent.
6-6
Yes, if the cash position of the firm is poor and if it has limited access to additional outside
financing. But even here, the relationship between present value and cost would be a better decision
tool.
6-7
a.
In general, the answer is no. The objective of management should be to maximize value, and
as we point out in subsequent chapters, stock values are determined by both earnings and
growth. The NPV calculation automatically takes this into account, and if the NPV of a
long-term project exceeds that of a short-term project, the higher future growth from the
long-term project must be more than enough to compensate for the lower earnings in early
years.
b.
If the same $100 million had been spent on a short-term project—one with a faster
payback—reported profits would have been higher for a period of years. Of course, this is
another reason why firms sometimes use the payback method.
____________________________________________________________
105
SOLUTIONS
6-1
a.
PB = $52,125/$12,000 = 4.34, so the payback is about 4 years.
b.
Numerical solution:
1
1  (1.12

)8


NPV  $52,125  $12,000
 0.12 
 $52,125  $12,000 (4.96764 )  $7,486 .68
Financial calculator Solution: Input the appropriate cash flows into the cash flow register,
input I = 12, and then solve for NPV = $7,486.68.
Spreadsheet Solution: Be careful when using the NPV function that is available with Excel,
because this function computes the present value of a series of nonconstant future cash flows
only. If you highlight a series of cash flows, the NPV function assumes the first cash flow is
CF1, not CF0. As a result, you should use the NPV function to compute the present value of
the future cash flows and then subtract the cash flow in the current period, which is the net
investment. For the current problem, you can use the PV function to compute the present
value of the $12,000 annuity and then subtract the $52,125 cost.
c.
Let NPV = 0. Therefore,
1
1  (1 IRR

)8
0
NPV  $52,125  $12,000 
 IRR 
1
1  (1 IRR

)8

$52,125  $12,000 
 IRR 
Numerical Solution: Without a financial calculator, you must use a trial-and-error process—
plug in values for IRR until the right side of the computation equals $52,125.
Financial calculator: Input the appropriate cash flows into the cash flow register and then
solve for IRR = 16%.
Spreadsheet solution: Because this is an annuity, you can use the RATE function that is
available on the spreadsheet. Pmt = $12,000 and Pv = -$52,125.
d.
Project L’s discounted payback period is calculated as follows:
106
Period
0
1
2
3
4
5
6
7
8
Cash Flows
($52,125)
12,000
12,000
12,000
12,000
12,000
12,000
12,000
12,000
Annual
Discounted @12%
Cash Flows
Cumulative
($52,125.00)
($52,125.00)
10,714.80
( 41,410.20)
9,566.40
( 31,843.80)
8,541.60
( 23,302.20)
7,626.00
( 15,676.20)
6,808.80
( 8,867.40)
6,079.20
( 2,788.20)
5,427.60
2,639.40
4,846.80
7,486.20
Discounted
$2,788.20
 6
 6.51 years
Payback
$5,427.60
6-2
a.
Compute the values for NPV at different required rates of return, k, by entering the cash flows
for each project in the CF register of your calculator, then change the value for I and compute
the NPV each time.
k
NPVA
NPVB
0%
10
12
15
18
20
24
30
$970
297
207
91
( 5)
( 60)
(152)
(256)
$399
179
146
102
64
41
0
( 51)
If you use the Excel spreadsheet to compute NPV, be careful, because this function computes
the present value of a series of future cash flows only. If you highlight a series of cash flows
contained in the spreadsheet, the NPV function assumes the first cash flow is CF1, not CF0. As
a result, you should use the NPV function to compute the present value of the future cash
flows and then subtract (add the negative) the cash flow in the current period, which is the net
investment. For the Project A, you might set up the spreadsheet as follows to compute net
present value at a 15 percent required rate of return:
107
Using the spreadsheet NPV function will compute the sum of the present values of CF1, CF2,
…, CF7:
Note the description of the spreadsheet NPV function given below the table. The range
entered into Value1 corresponds to CF1, CF2, …, CF7 in the spreadsheet setup. To compute the
net present value we describe in the book, we need to add the initial investment of -$300 to the
result given by the spreadsheet NPV function, which is $391.33. Therefore, net present value
we want to compute is $91.33 = -$300 + $391.33 at 15 percent:
108
To compute NPVs for various required rates of return, simply change the value for the
required rate of return in the spreadsheet (cell D1), and the result will be displayed
immediately.
If you set up your spreadsheet so that the NPVs for a series of required rates of return are
displayed sequentially either in a single column or in a single row, you can use the graph
function to draw the NPV profiles, which would look like the following:
NPV ($ millions)
1.0
0.8
Project A
0.6
0.4
Crossover
0.2
Project B
0
4
8
12 14.5 16 17.8
-0.2
109
k%
20
24
28
b.
For Project A, solve the following:
  $387    $193    $100   $500 
NPVA  $300  



1
2
3
4
 (1  IRR)   (1  IRR)   (1  IRR)   (1  IRR) 
 $500   $850   $100 



0
5
6
7 
 (1  IRR)   (1  IRR)   (1  IRR) 
Numerical Solution: Without a financial calculator, you must use a trial-and-error process—
plug in values for IRR until NPV = 0. Eventually you will find IRRA = 17.84% and IRRB =
23.97%.
Financial calculator: Input the appropriate cash flows into the cash flow register and then
solve for IRRA = 17.84% (IRRB = 23.97%).
Spreadsheet solution: Use the spreadsheet function called IRR. For Project A, we would have
the following:
110
In this case, note that the series of cells in the row labeled “Values” includes the cell that
contains the initial investment—that is CF0. The result given at the bottom of the table
indicates that Project A has an IRR equal to 17.84 percent:
c.
At k = 12%, Project A has the greater NPV, specifically $206.77 as compared to Project B’s
NPV of $145.93. Thus, Project A would be selected. At k = 15%, Project B has an NPV of
$102.12, which is higher than Project A’s NPV of $91.33. Thus, choose Project B if k = 15%.
d.
Looking at the NPV profile developed in Part a, the crossover rate is between 14 and 15
percent, or approximately 14.5%. To determine exactly where the crossover rate is, proceed as
follows: Construct a Project Δ, which is the difference in the two projects’ cash flows:
111
Year
1
2
3
4
5
6
7
Project Δ =
CFA – CFB
(521)
(327)
(234)
366
366
716
100
IRR = Crossover rate = 14.50%.
Projects A and B are mutually exclusive, thus, only one of the projects can be chosen. As long as
the required rate of return is greater than the crossover rate, both the NPV and IRR methods will lead
to the same project selection. However, if the required rate of return is less than the crossover rate
the two methods lead to different project selections—a conflict exists. When a conflict exists the
NPV method must be used.
6-3
a.
k
0
5
10
12
15
17
20
22
25
30
NPVA
2,400,000
1,714,286
1,090,909
857,143
521,739
307,692
0
( 196,721)
( 480,000)
( 923,077)
NPVB
30,000,000
14,170,642
5,878,484
3,685,832
1,144,596
( 181,689)
(1,773,883)
(2,633,441)
(3,696,845)
(5,036,832)
These NPVs can be computed using a calculator or a spreadsheet as described in the solution
to Problem 6-2a.
The crossover rate is approximately 16 percent. If the required rate of return is less than the
crossover rate, then Plan B should be accepted; if the required rate of return is greater than the
crossover rate, then Plan A is preferred. At the crossover rate, the two project NPVs are equal.
Thus, other criteria such as the IRR must be used to evaluate the projects. Using the same
procedure as in Problem 6-2d, the exact crossover rate is 16.07 percent.
112
NPV
(Millions of
Dollars)
30
Plan B
24
18
12
Crossover rate = 16.07%
6
Plan A
IRRA = 20.0%
2.4
k (%)
0
5
15
10
20
25
IRRB = 16.7%
6-4
b.
Yes. Assuming (1) equal risk among projects, and (2) that the required rate of return is a
constant and does not vary with the amount of funds raised, the firm would take on all
available independent projects with returns greater than its 12 percent required rate of return.
If the firm had invested in all available projects with returns greater than 12 percent, then its
best alternative would be to repay capital. Thus, the required rate of return is the correct
reinvestment rate for evaluating a project’s cash flows.
a.
Using a financial calculator or a spreadsheet (see the solutions to Problem 6-2a and b), we get:
NPVA = $14,486,808.
NPVB = $11,156,893.
IRRA = 15.03%.
IRRB = 22.26%.
113
b.
NPV
(Millions of
Dollars)
80
Plan A
60
40
Crossover rate = 11.7%
20 Plan B
IRRB = 22.26%
0
k (%)
5
10
15
20
25
IRRA = 15.03%
-20
The crossover rate is somewhere between 11 percent and 12 percent. Using the procedure
shown in 6-2 (d), the exact crossover rate is calculated as 11.7 percent.
c.
The NPV method implicitly assumes that the opportunity exists to reinvest the cash flows
generated by a project at the required rate of return, whereas use of the IRR method implies
the opportunity to reinvest at the IRR. All independent projects with an NPV > $0 should be
invested in by the firm. As cash flows come in from these projects, the firm will either pay
them out to investors, or use them as a substitute for outside funds that, in this case, costs 10
percent. Thus, because these cash flows are expected to save the firm 10 percent, this is their
opportunity cost reinvestment rate.
The IRR method assumes reinvestment at the internal rate of return itself, which is an
incorrect assumption, given a constant expected future required rate of return, and ready
access to capital markets.
6-5
See the solutions for Problem 6-2 for examples as to how to compute NPV and IRR using a
spreadsheet.
Pulley System:
1
1  (1.14

)5
  $17,100  $5,100 (3.43308 )
NPV  $17,100  $5,100 
 0.14 


 $17,100  $17,508 .71  $408 .71
The Pulley System project is acceptable.
Financial calculator: Input the appropriate cash flows into the cash flow register, input I = 14, and
then solve for NPV = $408.71.
IRR  15%.
(Accept)
114
Financial calculator: Input the appropriate cash flows into the cash flow register and then solve for
IRR = 14.99%  15%.
Payback period = PBPulley = $17,100/$5,100 = 3.35 years
Truck:
1
1  (1.14

)5
  $22,430  $7,500 (3.43308 )
NPV  $22,430  $7,500 
 0.14 


 $22,430  $25,748 .10  $3,318 .10
The Truck Project is acceptable.
Financial calculator: Input the appropriate cash flows into the cash flow register, input I = 14, and
then solve for NPV = $3,318.10.
IRR = 20%. (Accept)
Financial calculator: Input the appropriate cash flows into the cash flow register and then solve for
IRR = 20%.
Payback period = PBTruck = $22,430/$7,500 = 2.99  3 years
6-6
See the solutions for Problem 6-2 for examples as to how to compute NPV and IRR using a
spreadsheet.
Electric-powered:
1
1  (1.12

)6
  $22,000  $6,290 (4.11141)  $3,860 .75
NPVE  $22,000  $6,290 
 0.12 


Financial calculator: Input the appropriate cash flows into the cash flow register, input I = 12, and
then solve for NPV = $3,860.75.
IRRE = 18%.
Financial calculator: Input the appropriate cash flows into the cash flow register and then solve for
IRR = 18%.
Gas-powered:
1
1  (1.12

)6
  $17,500  $5,000 (4.11141)  $3,057 .05
NPVG  $17,500  $5,000 
 0.12 


115
Financial calculator: Input the appropriate cash flows into the cash flow register, input I = 12, and
then solve for NPV = $3,057.05.
IRRG = 18%.
Financial calculator: Input the appropriate cash flows into the cash flow register and then solve for
IRR = 17.97% ≈ 18%.
Horrigan Industries should purchase the electric-powered forklift because it has a higher NPV than
the gas-powered forklift. The company gets a high rate of return (18% > k = 12%) on a larger
investment.
6-7
See the solutions for Problem 6-2 for examples as to how to compute NPV and IRR using a
spreadsheet.
a.
Using a financial calculator:
NPVS = $448.86
NPVL = $607.20
IRRS = 15.24%
IRRL = 14.67%
PBS =
6-8
3.3 yrs
PBL =
3.4 yrs
b.
If the projects are independent, both are acceptable—both NPVS and NPVL are positive. If the
projects are mutually exclusive, Project L should be chosen because NPVL > NPVS.
a.
The PV of costs for the conveyor system is –$556,717, while the PV of costs for the forklift
system is –$493,407. Thus, the forklift system is expected to be –$493,407 – (–$556,717) =
$63,310 less costly than the conveyor system, and hence the forklifts should be used.
1
1  (1.09

)5

PV costs of conveyor  300,000  (66,000) 
 0.09 


 300,000  (66,000 )(3.88965 )  556,716 .90
b.
6-9
The IRRs of the two alternatives are undefined. To calculate an IRR, the cash flow stream
must include both cash inflows and outflows.
See the solutions for Problem 6-2 for examples as to how to compute NPV and IRR using a
spreadsheet.
NPVC  $14,000 
$8,000 $6,000 $2,000 $3,000



(1.12)1 (1.12) 2 (1.12) 3 (1.12) 4
 $14,000  $8,000 (0.89286 )  $6,000 (0.79719 )
 $2,000 (0.71178 )  $3,000 (0.63552 )
116
 $14,000  $7,142 .88  $4,783 .14  $1,423 .56  $1,906 .56
 $14,000  $15,256 .14
 $1,256 .14
Calculator solution: NPVC = $1,256.14
IRRC = 17.3%
1
1  (1.12

)4
  $22,840  $8,000 (3.03735 )  $1,458 .80
NPVR  $22,840  $8,000 
 0.12 


Calculator solution: NPVR = $1,458.79
IRRR = 15.0%
NPVR > NPVC, so Project R should be accepted.
6-10
See the solutions for Problem 6-2 for examples as to how to compute NPV and IRR using a
spreadsheet.
$2,000
$900
$100
$100
NPVD  $2,500 



1
2
3
(1.14)
(1.14)
(1.14)
(1.14) 4
 $2,500  $2,000 (0.87719 )  $900 (0.76947 )  $100 (0.67497 )  $100 (0.59208 )
 $2,500  $1,754 .38  $692 .52  $67.50  $59.21
 $2,500  $2573 .71  $73.61
Calculator solution: NPVD = $73.61
IRRD = 16.4%
NPVQ  $2,500 
$0
$1,800 $1,000
$900



1
2
3
(1.14)
(1.14)
(1.14)
(1.14) 4
 $2,500  $0(0.87719 )  $1,800 (0.76947 )  $1,000 (0.67497 )  $900 (0.59208 )
 $2,500  $0  $1,385 .05  $674 .97  $532 .87
 $2,500  $2592 .89  $92.89
Calculator solution: NPVQ = $92.89
IRRQ = 15.6%
NPVQ > NPVD, so Project Q is the better project, and its IRR is 15.6%.
6-11
See the solutions for Problem 6-2 for examples as to how to compute NPV and IRR using a
spreadsheet.
117
NPVY  $25,000 
$10,000 $9,000 $7,000 $6,000



(1.10)1 (1.10) 2 (1.10) 3 (1.10) 4
 $25,000  $10,000 (0.90909 )  $9,000 (0.82645 )
 $7,000 (0.75131)  $6,000 (0.68301)
 $25,000  $9,090 .90  $7,438 .05  $5,259 .17  $4,098 .06
 $25,000  $25,886 .18  $886 .18
Calculator solution: NPVY = $886.21
IRRY = 11.8%
NPVz  $25,000 
$0
$0
$0
$36,000



1
2
3
(1.10)
(1.10)
(1.10)
(1.10) 4
 $25,000  $0(0.90909)  $0(0.82645)  $0(0.75131)  $36,000(0.68301)
 $25,000  $0  $0  $0  $24,588.36  $25,000  $24,588.36  $411.64
Calculator solution: NPVY = -$411.51
IRRZ = 9.5%
NPVY > 0 and NPVZ < 0, so only Project Y should be accepted.
6-12
6-13
The solution is given in the Instructor's Manual, Solutions to Integrative Problems.
Computer-Related Problem
a.
The NPV profiles are shown next. These profiles were created using the computerized
model C06 and the data used to construct these profiles are given after the graph.
118
NPV Profiles
NPV ($ thousands)
50
40
Project A
30
20
10
0
Project B
4
8
12
16
20
24
28
k (%)
-10
-20
-30
Data for NPV profile graph:
Cost of
Capital
0%
2%
4%
6%
8%
10%
12%
14%
16%
18%
20%
22%
24%
26%
28%
30%
b.
Proj. A
NPV
$41,000
33,285
26,350
20,104
14,466
9,367
4,747
554
(3,260)
(6,733)
(9,902)
(12,797)
(15,446)
(17,874)
(20,102)
(22,149)
Proj. B
NPV
$25,000
20,702
16,777
13,185
9,891
6,862
4,072
1,496
(886)
(3,092)
(5,141)
(7,045)
(8,819)
(10,474)
(12,020)
(13,466)
For Project A, IRR = 14.28%. For Project B, IRR = 15.24%.
INPUT DATA:
Year
0
1
2
3
4
5
KEY OUTPUT:
Expected
Cash Flows
Proj. A
Proj. B
($45,000)($50,000)
(20,000) 15,000
11,000
15,000
20,000
15,000
30,000
15,000
45,000
15,000
Required rate of return
NPV
IRR
Proj. A
$2,600
14.28%
Crossover rate
13.00%
119
Proj. B
$2,758
15.24%
12.81%
c.
At a required rate of return of 13 percent, NPVA = $2,600, while NPVB = $2,758.
Therefore, Project B should be selected. From the data shown below, you can see that
Project A is preferable if the required rate of return is 9 percent (NPVA = $11,853, NPVB =
$8,345). At a required rate of return of 15 percent, Project B is preferable (NPVA =
($1,398), NPVB = $282).
INPUT DATA:
Year
0
1
2
3
4
5
KEY OUTPUT:
Expected Cash Flows
Proj. A
Proj. B
($45,000)
($50,000)
( 20,000)
15,000
11,000
15,000
20,000
15,000
30,000
15,000
45,000
15,000
Required rate of return
INPUT DATA:
NPV
IRR
Crossover rate
Proj. B
$8,345
15.24%
12.81%
9.00%
KEY OUTPUT:
Expected Cash Flows
Year Proj. A
Proj. B
0 ($45,000)
($50,000)
1 ( 20,000)
15,000
2
11,000
15,000
3
20,000
15,000
4
30,000
15,000
5
45,000
15,000
NPV
IRR
Proj. A
($1,398)
14.28%
Crossover rate
Required rate of return
d.
Proj. A
$11,853
14.28%
Proj. B
$282
15.24%
12.81%
15.00%
The NPV profiles cross at 12.81%. The calculations used to determine this rate are
given below:
INPUT DATA:
Expected Cash Flows
Year Proj. A
Proj. B
0 ($45,000) ($50,000)
1
(20,000)
15,000
2
11,000
15,000
3
20,000
15,000
4
30,000
15,000
5
45,000
15,000
Required rate of return
13.00%
Crossover rate calculation:
Delta’s
Year
CFs
0
$5,000
1
(35,000)
2
( 4,000)
3
5,000
4
15,000
5
30,000
NPV Delta ($ 180)
IRR Delta
12.81%
Required
Return
0.00%
5.00%
10.00%
11.12%
20.00%
30.00%
NPV of
Delta
$16,000
8,204
2,505
1,450
(4,761)
(8,682)
120
e.
Worst case scenario: Year 5 Cash flow = $40,000
INPUT DATA:
KEY OUTPUT:
Expected
Cash Flows
Year Proj. A
Proj. B
0
($45,000) ($50,000)
1
( 20,000)
15,000
2
11,000
15,000
3
20,000
15,000
4
30,000
15,000
5
40,000
15,000
Proj. A
($114)
12.94%
NPV
IRR
Proj. B
$2,758
15.24%
Crossover rate
Required rate of return
9.30%
13.00%
BEST CASE SCENARIO: Year 5 cash flow = $50,000
INPUT DATA:
KEY OUTPUT:
Expected
Year Proj. A
0 ($45,000)
1 ( 20,000)
2
11,000
3
20,000
4
30,000
5
50,000
Cash Flows
Proj. B
($50,000)
15,000
15,000
15,000
15,000
15,000
Proj. A
$5,314
15.55%
NPV
IRR
Crossover rate
Required rate of return
Proj. B
$2,758
15.24%
16.01%
13.00%
Solutions to Appendix Problem
6B-1
a.
The project’s expected cash flows are as follows (in millions of dollars):
Time
0
1
2
Net Cash Flow
$( 2.0)
13.0
(12.0)
We can construct the following NPV profile:
NPV
(Millions of
Dollars)
1.5
1
0.5
0
-0.5
-1
0
Discount Rate
0%
100
200
300
NPV
$(1,000,000)
121
400
500
Discount
rate (%)
10
50
80
100
200
300
400
410
420
430
450
( 99,174)
1,333,333
1,518,519
1,500,000
1,000,000
500,000
120,000
87,659
56,213
25,632
( 33,058)
b.
If k = 10%, reject the project because NPV < $0. Its NPV at k = 10% is equal to -$99,174.
But if k = 20%, accept the project because NPV > $0. Its NPV at k = 20% is $500,000.
c.
Other possible projects with multiple rates of return could be nuclear power plants where
disposal of radioactive wastes is required at the end of the project's life.
d.
MIRR @ k = 10%:
PV costs = $2,000,000 + $12,000,000/(1.10) 2 = $11,917,355.
FV inflows = $13,000,000 H 1.10 = $14,300,000.
MIRR = 9.54%. (Reject the project since MIRR < k.)
MIRR @ k = 20%:
PV costs = $2,000,000 + $12,000,000/(1.20) 2 = $10,333,333.
FV inflows = $13,000,000 H 1.20 = $15,600,000.
MIRR = 22.87%. (Accept the project since MIRR > k.)
Looking at the results, the MIRR calculations would lead you to the same decisions as the
NPV calculations did.
122