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Unformatted text preview: Chapter 6: Some Alternative Investment Rules
6.1 a. The payback period is the time that it takes for the cumulative undiscounted cash inflows
to equal the initial investment.
Project A:
Cumulative Undiscounted Cash Flows Year 1
Cumulative Undiscounted Cash Flows Year 2
Payback period = $4,000
= $4,000 +$3,500 = $4,000
= $7,500 =2 Project A has a payback period of two years.
Project B:
Cumulative Undiscounted Cash Flows Year 1
Cumulative Undiscounted Cash Flows Year 2
Cumulative Undiscounted Cash Flows Year 3 = $2,500
= $2,500
= $2,500+$1,200
= $3,700
= $2,500+$1,200+$3,000 = $6,700 Project B’s cumulative undiscounted cash flows exceed the initial investment of $5,000
by the end of year 3. Many companies analyze the payback period in whole years. The
payback period for project B is 3 years.
Project B has a payback period of three years.
Companies can calculate a more precise value using fractional years. To calculate the
fractional payback period, find the fraction of year 3’s cash flows that is needed for the
company to have cumulative undiscounted cash flows of $5,000. Divide the difference
between the initial investment and the cumulative undiscounted cash flows as of year 2
by the undiscounted cash flow of year 3.
Payback period = 2 + ($5,000  $3,700) / $3,000
= 2.43 Since project A has a shorter payback period than project B has, the company
should choose project A.
b. Discount each project’s cash flows at 15 percent. Choose the project with the highest
NPV.
Project A = $7,500 + $4,000 / (1.15) + $3,500 / (1.15)2 + $1,500 / (1.15)3
= $388.96 Project B = $5,000 + $2,500 / (1.15) + $1,200 / (1.15)2 + $3,000 / (1.15)3
= $53.83 The firm should choose Project B since it has a higher NPV than Project A has.
6.2 a. Find the payback period for the project. Since the cash inflows are constant,
divide the initial investment by the annual cash inflow to determine the payback period.
Payback Period = Initial Investment / Annual Cash Inflow
= $1,000,000 / $150,000
= 6.67 B92 The payback period is 6.67 years. Since the payback period is shorter than the
cutoff period of ten years, the project should be accepted.
b. Find the number of years needed for the discounted cash inflows to equal the initial
investment of $1 million. Apply the annuity formula, discounted at 10 percent, to find
the approximate discounted payback period. The approximate discounted payback period
is the year in which the PV of the initial investment is surpassed.
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 A70.1
= $730,262.82
Cumulative Discounted Cash Flows Year 8 = $150,000 A80.1
= $800,238.93
Cumulative Discounted Cash Flows Year 9 = $150,000 A90.1
= $863,853.57
Cumulative Discounted Cash Flows Year 10 = $150,000 A100.1 = $921,685.07
Cumulative Discounted Cash Flows Year 11 = $150,000 A110.1 = $974,259.15
Cumulative Discounted Cash Flows Year 12 = $150,000 A120.1 = $1,022,053.77
The cumulative discounted cash flows exceed the initial investment of $1,000,000 by the
end of year 12. Many companies analyze the payback period in whole years. 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.
6.3 a. The average accounting return is the average project earnings after taxes, divided by the
average book value, or average net investment, of the machine during its life. The book
value of the machine is the gross investment minus the accumulated depreciation.
Average Book Value = (Book Value0 + Book Value1 + Book Value2 + Book Value3 +
Book Value4 + Book Value5) / (Economic Life)
= ($16,000 + $12,000 + $8,000 + $4,000 + $0) / (5 years)
= $8,000 Average Project Earnings = $4,500
Divide the average project earnings by the average book value of the machine to calculate
the average accounting return.
Average Accounting Return = Average Project Earnings / Average Book Value
= $4,500 / $8,000
= 0.5625
= 56.25% The average accounting return is 56.25%. B93 b. The average accounting return uses accounting data rather than net cash flows. 2. The average accounting return uses an arbitrary firm standard as the decision
rule. The firm standard is arbitrary because it does not necessarily relate to a
market rate of return. 3.
6.4 1. The average accounting return does not consider the timing of cash flows.
Hence, it does not consider the time value of money. Determine the average book value of the investment. The book value is the gross investment
minus accumulated depreciation. Gross Investment
Less: Accumulated
Depreciation
Net Investment Purchase
$2,000,000 Year 1
$2,000,000 Year 2
$2,000,000 Year 3
$2,000,000 Year 4
$2,000,000 Year 0
$2,000,000 0
$2,000,000 400,000
$1,600,000 800,000
$1,200,000 1,200,000
$800,000 1,600,000
$400,000 2,000,000
$0 Average Book Value = ($2,000,000 + $1,600,000 + $1,200,000 + $800,000
+ $400,000 + $0) / (6)
= $1,000,000 Next, calculate average annual net income.
Net Income Year 1
Net Income Year 2
Net Income Year 3
Net Income Year 4
Net Income Year 5 = $100,000
= $100,000 (1.07)
= $100,000 (1.07)2
= $100,000 (1.07)3
= $100,000 (1.07)4 Average Net Income = ($100,000+$107,000+$114,490+$122,504+$131,080) / 5
= $115,015 = $107,000
= $114,490
= $122,504
= $131,080 The average accounting return is the average net income divided by the average book value.
Average Accounting Return = Average Net Income / Average Book Value
= $115,015 / $1,000,000
= 0.115
= 11.5% Since the machine’s average accounting return, 11.5%, is below the company’s cutoff of
20%, the machine should not be purchased.
6.5 First determine the average book value of the project. The book value is the gross investment
minus accumulated depreciation.
Gross Investment
Less: Accumulated
Depreciation
Net Investment Purchase Date
$8,000 Year 1
$8,000 Year 2
$8,000 Year 3
$8,000 0
$8,000 4,000
$4,000 6,500
$1,500 8,000
$0 B94 Average Book Value = ($8,000 + $4,000 + $1,500 + $0) / (4 years)
= $3,375
Remember to use the aftertax average net income when calculating the average accounting return.
Average Aftertax Net Income = (1 – Tc) Annual Pretax Net Income
= (1 – 0.25) $2,000
= $1,500 The average accounting return is the average aftertax net income divided by the average book
value.
Average Accounting Return = $1,500 / $3,375
= 0.44
= 44% The average accounting return of the machine is 44%.
6.6 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)
0
IRR = C0 + C1 / (1+IRR) + C2 / (1+IRR)2
= $3,000 + $2,500 / (1+IRR) + $1,000 / (1+IRR)2
= 0.1289 IRR(Project B)
0
IRR = C0 + C1 / (1+IRR) + C2 / (1+IRR)2
= $6,000 + $5,000 / (1+IRR) + $1,000 / (1+IRR)2
= 0.1289 Note that since Project B’s cash flows are two times those of Project A, the IRR’s of both projects
are the same.
The IRR of both Project A and Project B is 12.89%.
6.7 a. The internal rate of return is the discount rate at which the NPV of the project’s cash
flows equal zero. Set the project’s cash flows, discounted at the internal rate of return
(IRR), equal to zero. Solve for the IRR.
IRR
0
IRR = C0 + C1 / (1+IRR) + C2 / (1+IRR)2 + C3 / (1+IRR)3
= $8,000 + $4,000 / (1+IRR) + $3,000 / (1+IRR)2 + $2,000 / (1+IRR)3
= 0.0693 The IRR is 6.93%.
b. No. An investingtype project is one with a negative initial cash outflow and positive
future cash inflows. One accepts a project when the IRR is greater than the discount rate.
Similarly, one rejects the project when the IRR is less than the discount rate. The project
should not be accepted because the IRR (6.93%) is less than the discount rate (8%). B95 6.8 Set the project’s cash flows, discounted at the internal rate of return (IRR), equal to zero. Solve
for the IRR.
IRR(Project A)
0
IRR = C0 + C1 / (1+IRR) + C2 / (1+IRR)2 + C3 / (1+IRR)3
= $2,000 + $2,000 / (1+IRR) + $8,000 / (1+IRR)2 + $8,000 / (1+IRR)3
= 1.88 IRR(Project B)
0
IRR = C0 + C1 / (1+IRR) + C2 / (1+IRR)2 + C3 / (1+IRR)3
= $1,500 + $500 / (1+IRR) + $1,000 / (1+IRR)2 + $1,500 / (1+IRR)3
= 0.362 The IRR for Project A is 188% and the IRR for Project B is 36.2%.
6.9 a. Set the project’s cash flows, discounted at the internal rate of return (IRR), equal to zero.
Solve for the IRR.
IRR
0
IRR = C0 + C1 / (1+IRR) + C2 / (1+IRR)2 + C3 / (1+IRR)3 + C4 / (1+IRR)4
= $5,000  $2,500 / (1+IRR)  $2,000 / (1+IRR)2  $1,000 / (1+IRR)3
 $1,000 / (1+IRR)4
= 0.1399 The IRR is 13.99%.
b. This problem differs from previous ones because the initial cash flow is positive and all
future cash flows are negative. In other words, this is a financingtype project while
previous projects were investingtype projects. For financing situations, accept the
project when the IRR is less than the discount rate. Reject the project when the IRR is
greater than the discount rate.
IRR
Discount Rate = 13.99%
= 10% IRR > Discount Rate
Reject the offer when the discount rate is less than the IRR.
c. IRR
Discount Rate = 13.99%
= 20% IRR < Discount Rate
Accept the offer when the discount rate is greater than the IRR.
d. Calculate the NPV when the discount rate is 10 percent.
NPV = $5,000  $2,500 / (1.1)  $2,000 / (1.1)2  $1,000 / (1.1)3  $1,000 / (1.1)4
= $359.95 When the discount rate is 10 percent, the NPV of the offer is $359.95. Reject the
offer.
Calculate the NPV when the discount rate is 20 percent.
NPV = $5,000  $2,500 / (1.2)  $2,000 / (1.2)2  $1,000 / (1.2)3  $1,000 / (1.2)4
= $466.82
B96 When the discount rate is 20 percent, the NPV of the offer is $466.82. Accept the
offer.
e.
6.10 Yes, the decisions under the NPV rule are consistent with the choices made under
the IRR rule since the signs of the cash flows change only once. a. Set the project’s cash flows, discounted at the internal rate of return (IRR), equal to zero.
Solve for the IRR.
IRR(Project A)
0
IRR = C0 + C1 / (1+IRR) + C2 / (1+IRR)2
= $5,000 + $3,500 / (1+IRR) + $3,500 / (1+IRR)2
= 0.2569 The IRR of project A is 25.69%.
IRR(Project B)
0
IRR = C0 + C1 / (1+IRR) + C2 / (1+IRR)2
= $100,000 + $65,000 / (1+IRR) + $65,000 / (1+IRR)2
= 0.1943 The IRR of project B is 19.43%.
b. Choose project A because it has a higher IRR than project B. c. The difference in scale was ignored. Project B has a substantially larger initial
investment than project A has. Thus, the simple IRR calculation may not lead to the best
decision. d. Calculate the incremental IRR. The incremental IRR is the IRR on the incremental
investment from choosing the larger project instead of the smaller project. The
incremental cash flows are the differences between the cash flows of project B and those
of project A. Always subtract the project with the smaller initial cash outflow from the
project with the larger initial cash outflow. In this way, the initial incremental cash flow
will be negative.
Project B Cash Flows
Project A Cash Flows
B –A Year 0
$100,000
5,000
$95,000 Year 1
$65,000
3,500
$61,500 Year 2
$65,000
3,500
$61,500 Next, find the IRR of those incremental cash flows.
IRR(B – A)
0
IRR = C0 + C1 / (1+IRR) + C2 / (1+IRR)2
= $95,000 + $61,500 / (1+IRR) + $61,500 / (1+IRR)2
= 0.191 The incremental IRR is 19.1%.
e. For investingtype projects, accept the larger project when the incremental rate of return
is greater than the discount rate. Therefore, choose project B since the incremental IRR
(19.1%) is greater than the 15 percent discount rate. B97 f. Calculate the NPV of each project.
NPV(Project A) = $5,000 + $3,500 / (1.15) + $3,500 / (1.15)2
= $689.98
The NPV of project A is $689.98.
NPV(Project B) = $100,000 + $65,000 / (1.15) + $65,000 / (1.15)2
= $5,671.08
The NPV of project B is $5,671.08.
Since the NPV of project B, $5,671.08, is greater than the NPV of project A, $689.98,
choose project B. 6.11 a. Apply the growing perpetuity formula to find the PV of stream A. The perpetuity
formula values the stream as of one year before the first payment. Therefore, the
growing perpetuity formula values the stream of cash flows as of year 2. Next, discount
the PV as of the end of year 2 back two years to find the PV as of today, year 0.
PV(A) = [C3 / (r – g)] / (1+r)2
= [$5,000 / (0.12  0.04)] / (1.12)2
= $49,824.62 The PV of stream A is $49,824.62.
Apply the perpetuity formula to find the PV of stream B. The perpetuity formula
discounts the stream back to year 1, one period prior to the first cash flow. Discount the
PV as of the end of year 1 back one year to find the PV as of today, year 0.
PV(B) = [C2 / (r)] / (1+r)
= [$6,000 / (0.12)] / (1.12)
= $44,642.86 The PV of stream B is $44,642.86.
b. Streams A and B are combined to form project C.
Project A = [C3 / (r – g)] / (1+r)2 Project B = [C2 / (r)] / (1+r) Project C = Project A + Project B
= [C3 / (r – g)] / (1+r)2 + [C2 / (r)] / (1+r) Set the new project’s cash flows, discounted at the internal rate of return (IRR), equal to
zero. Solve for the IRR. Use a graphing calculator to perform the calculation.
IRR(Project C)
0
IRR = [C3 / (IRR – g)] / (1+IRR)2 + [C2 / (IRR)] / (1+IRR)
= [$5,000 / (IRR  0.04)] / (1+IRR)2 + [$6,000 / (IRR)] / (1+IRR)
= 0.1465 The IRR for project C is 14.65%. B98 c. 6.12 The correct decision rule for an investingtype project is to accept the project if the
discount rate is below the IRR. Since there is one IRR, a decision can be made. At a
point in the future, the cash flows from stream A will be greater than those from stream B.
Therefore, although there are many cash flows, there will be only one change in sign.
When the sign of the cash flows change more than once over the life of the project, there
may be multiple internal rates of return. In such cases, there is no correct decision rule
for accepting and rejecting projects using the internal rate of return. False. The statement is false. If the cash flows of project B occur early and the cash flows of
project A occur late, then for a low discount rate the NPV of A can exceed the NPV of B. Observe
the following example.
Project A
Project B C0
$1,000,000
2,000,000 C1
$0
2,400,000 C2
$1,440,000
0 IRR
0.20
0.20 NPV @ 0%
$440,000
400,000 However, in one particular case, the statement is true for equally risky projects. If the lives of the
two projects are equal and the cash flows of project B are twice the cash flows of project A in
every time period, the NPV of project B will be twice the NPV of project A.
6.13 a. The profitability index, PI, is the ratio of the present value of the future expected cash
flows after the initial investment to the amount of the initial investment.
PI(A) = [C1 / (1+r) + C2 / (1+r)2 + C3 / (1+r)3] / (Initial Investment)
= [$300 / (1.1) + $700 / (1.1)2 + $600 / (1.1)3] / ($500)
= 2.6 The profitability index for project A is 2.6.
PI(B) = [C1 / (1+r) + C2 / (1+r)2 + C3 / (1+r)3] / (Initial Investment)
= [$300 / (1.1) + $1,800 / (1.1)2 + $1,700 / (1.1)3] / ($2,000)
= 1.5 The profitability index for project B is 1.5.
b. Greenplain should accept both projects A and B. The NPV of a project is positive
whenever the profitability index (PI) is greater than one. 6.14 Although the profitability index (PI) is higher for project B than for project A, project A should be
chosen because it has the greater NPV. Confusion arises because project B requires a smaller
investment than project A requires. Since the denominator of the PI ratio is lower for project B
than for project A, B can have a higher PI yet have a lower NPV. Only in the case of capital
rationing could Global Investments’ decision have been incorrect. 6.15 a. The profitability index, PI, is the ratio of the present value of the future expected cash
flows after the initial investment to the amount of the initial investment.
PI(A) = [C1 / (1+r) + C2 / (1+r)2] / (Initial Investment)
= [$70,000 / (1.12) + $70,000 / (1.12)2] / ($100,000)
= 1.183 The profitability index for project A is 1.183. B99 PI(B) = [C1 / (1+r) + C2 / (1+r)2] / (Initial Investment)
= [$130,000 / (1.12) + $130,000 / (1.12)2] / ($200,000)
= 1.099 The profitability index for project B is 1.099.
PI(C) = [C1 / (1+r) + C2 / (1+r)2] / (Initial Investment)
= [$75,000 / (1.12) + $60,000 / (1.12)2] / ($100,000)
= 1.148 The profitability index for project C is 1.148.
b. = $100,000 + $70,000 / (1.12) + $70,000 / (1.12)2
= $18,303.57 NPV (A) The NPV of project A is $18,303.57.
= $200,000 + $130,000 / (1.12) + $130,000 / (1.12)2
= $19,706.63 NPV(B) The NPV of project B is $19,706.63.
= $100,000 + $75,000 / (1.12) + $60,000 / (1.12)2
= $14,795.92 NPV(C) The NPV of project C is $14,795.92.
c. Accept projects A, B, and C. Since the projects are independent, accept all three
projects because their respective profitability indices are greater than one. 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 initial investment, the problem of scale
does not arise when comparing the profitability indices. Based on the profitability index
rule, project C can be eliminated because its PI is less than the PI of project A.
Because of the problem of scale, one cannot compare the PIs of projects A and
B. However, one can calculate the PI of the incremental cash flows of the two projects.
Project
B–A C0
$100,000 C1
$60,000 C2
$60,000 PI
1.014 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 PI calculation is:
PI(B – A) = [$60,000 / (1.12) + $60,000 / (1.12)2] / ($100,000)
= 1.014 Amaro should accept project B since the PI of the incremental cash flows is greater than
one.
e. Project B has the highest NPV. Project A has the next highest NPV. Therefore, accept
projects B and A.
B100 6.16 The PV of the cash inflows subsequent to the initial investment can be calculated using the sevenyear annuity formula, discounted at 15 percent. Divide the PV of those cash flows by the initial
investment to find the profitability index, PI. Bill should accept the project if the PI is greater than
one.
PI = [C1 ATr] / (Initial Investment)
= [$40,000 A70.15] / ($160,000)
= 1.04 Bill should accept the project since it has a profitability index greater than one.
6.17 a. The payback period is the time it takes to recoup the initial investment of a project.
Accept any project with a payback period equal to or shorter than the company’s standard
payback period. Reject all other projects. b. The average accounting return (AAR) is defined as the average project earnings divided
by the average book value of the investment. Accept projects for which the AAR is equal
to or greater than the firm’s standard average accounting return. Reject all other projects. c. The internal rate of return (IRR) is the discount rate that makes the net present value
(NPV) of a project equal to zero. The acceptance and rejection criteria are:
If C0 < 0 and all future cash flows are positive, accept the project if the internal rate of
return is greater than or equal to the discount rate.
If C0 < 0 and all future cash flows are positive, reject the project if the internal rate of
return is less than the discount rate.
If C0 > 0 and all future cash flows are negative, accept the project if the internal rate of
return is less than or equal to the discount rate.
If C0 > 0 and all future cash flows are negative, reject the project if the internal rate of
return is greater than the discount rate.
If the project has cash flows with more than one change in sign, there is likely to be more
than one positive IRR. In that situation, there is no valid IRR accept/reject rule. d. e. 6.18 The profitability index (PI) is the present value of the cash flows subsequent to the initial
investment divided by the initial investment. Accept any project for which the
profitability index is greater than or equal to one. Reject any project that has a PI less
than one.
The net present value (NPV) is the sum of the present values of all project cash flows.
Accept those projects with NPVs that are greater than or equal to zero. Reject projects
with negative NPVs. a. Payback period for the New Sunday Early Edition:
Use the payback period rule to calculate the number of years that it takes for the
cumulative undiscounted cash inflows to equal the initial investment.
Initial Investment
Year 1 = $600
Year 2 = $600 + $550
Year 3 = $600 + $550 + $450 = $1,200
= $600
= $1,150
= $1,600 B101 The undiscounted cash flows exceed the initial investment of $1,200 by the end of year 3.
Many companies analyze the payback period in whole years. The payback period for the
project is 3 years.
The New Sunday Early Edition has a payback period of three years.
Companies can calculate a more precise value using fractional years. Calculate the
fraction of year 3’s cash flow that is needed for the company to have cumulative
undiscounted cash flows of $1,200. Find the difference between the initial investment
and the cumulative undiscounted cash flows as of year 2, divided by the undiscounted
cash flow of year 3.
Payback Period = 2 + ($1,200  $1,150) / $450
= 2.11
Payback period for the New Saturday Late Edition:
Use the payback period rule to calculate the number of years that it takes for the
cumulative undiscounted cash inflows to equal the initial investment.
Initial Investment
Year 1 = $1,000
Year 2 = $1,000 + $900
Year 3 = $1,000 + $900 + $800 = $2,100
= $1,000
= $1,900
= $2,700 In year 3, the undiscounted cash flows exceed the initial investment of $2,100 by the end
of year 3. Many companies analyze the payback period in whole years. The payback
period for the project is 3 years.
The payback period for the New Saturday Late Edition is three years.
Companies can calculate a more precise value using fractional years. Calculate the
fraction of year 3’s cash flows that is needed for the company to have cumulative
undiscounted cash flows of $2,100. Find the difference between the initial investment
and the cumulative undiscounted cash flows as of year 2, divided by the undiscounted
cash flow of year 3.
Payback Period = 2 + ($2,100  $1,900) / $800
= 2.25
Using the whole number payback period, the projects are equally attractive. Using
the fractional payback period calculation, the New Sunday Early Edition is more
attractive because it has a shorter payback period than does the New Saturday
Early Edition.
b. New Sunday Early Edition IRR
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
0
IRR = C0 + C1 / (1+IRR) + C2 / (1+IRR)2 + C3 / (1+IRR)3
= $1,200 + $600 / (1+IRR) + $550 / (1+IRR)2 + $450 / (1+IRR)3
= 0.1676 New Saturday Late Edition IRR
B102 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
0
IRR = C0 + C1 / (1+IRR) + C2 / (1+IRR)2 + C3 / (1+IRR)3
= $2,100 + $1,000 / (1+IRR) + $900 / (1+IRR)2 + $800 / (1+IRR)3
= 0.1429 The New Sunday Early Edition has a greater IRR than the New Saturday Late
Edition.
c. Find the IRR of the incremental cash flows. The incremental IRR is the IRR on the
incremental investment from choosing the larger project instead of the smaller project.
Incremental cash flows are defined as the New Saturday Late Edition’s Cash Flows
minus the New Sunday Early Edition’s cash flows. Remember to subtract the cash flows
of the project with the smaller initial investment from those of the project with the larger
initial investment, so that the incremental initial investment is negative.
Year 0
$2,100
1,200
$900 Saturday Edition
Sunday Edition
Saturday – Sunday
IRR
0
IRR Year 1
$1,000
600
$400 Year 2
$900
550
$350 Year 3
$800
450
$350 = C0 + C1 / (1+IRR) + C2 / (1+IRR)2 + C3 / (1+IRR)3
= $900 + $400 / (1+IRR) + $350 / (1+IRR)2 + $350 / (1+IRR)3
= 0.1102 For investingtype projects, accept the larger project when the incremental rate of
return is greater than the discount rate. Since the discount rate of 12% is greater
than the incremental IRR of 11.02%, choose the new Sunday Edition.
d. Average Accounting Return for the New Sunday Early Edition:
First, determine the average book value of the project. The book value is the gross
investment minus accumulated depreciation.
Annual Depreciation = $1,200 / 3
= $400 Gross Investment
Accumulated Depreciation
Book Value
Average Investment Year 0
$1,200
$0
$1,200 Year 1
$1,200
$400
$800 = ($1,200 + $800 + $400 + $0) / (4)
= $600 Calculate the average annual income of the project.
Average Income = ($400 + $350 + $300) / 3
= $350 B103 Year 2
$1,200
$800
$400 Year 3
$1,200
$1,200
$0 Divide the average project earnings by the average book value of the machine to calculate
the average accounting return.
Average Accounting Return = (Average Income) / (Average Investment)
= $350 / $600
= 0.583 The average accounting return for the New Sunday Early Edition is 58.3%.
Average Accounting Return for the New Saturday Late Edition:
First, determine the average book value of the project. The book value is the gross
investment minus accumulated depreciation.
Annual Depreciation = $2,100 / 3
= $700 Gross Investment
Accumulated Depreciation
Book Value
Average Investment Year 0
$2,100
$0
$2,100 Year 1
$2,100
$700
$1,400 Year 2
$2,100
$1,400
$700 Year 3
$2,100
$2,100
$0 = ($2,100 + $1,400 + $700 + $0) / (4)
= $1,050 Calculate the average annual income of the project.
Average Income = ($800 + $700 + $600) / 3
= $700 Divide the average project earnings by the average book value of the machine to calculate
the average accounting return.
Average Accounting Return = (Average Income) / (Average Investment)
= $700 / $1,050
= 0.667 The average accounting return for the New Saturday Late Edition is 66.7%.
6.19 a. Discounted Payback Period for Deepwater Fishing:
Find the number of years that it takes for the discounted cash inflows to equal the initial
investment of $600,000. The approximate discounted payback period is the year in
which the PV of the initial investment is surpassed.
Cumulative Discounted Cash Flows Year 1 = $270,000 / (1.15)
= $234,782.61
Cumulative Discounted Cash Flows Year 2 = $270,000 / (1.15) + $350,000 / (1.15)2
= $499,432.89
Cumulative Discounted Cash Flows Year 3 = $270,000 / (1.15) + $350,000 / (1.15)2 +
$300,000 / (1.15)3
= $696,687.76 B104 The cumulative discounted cash flows exceed the initial investment of $600,000 by the
end of year 3. Many companies analyze the payback period in whole years. The payback
period for the project is three years.
The discounted payback period for deepwater fishing is three years.
Discounted Payback Period for New Submarine Ride:
Find the number of years that it will take the discounted cash inflows to equal the initial
investment of $1,800,000. The approximate discounted payback period is the year in
which the PV of the initial investment is surpassed.
Cumulative Discounted Cash Flows Year 1 = $1,000,000 / (1.15)
= $869,565.22
Cumulative Discounted Cash Flows Year 2 = $1,000,000 / (1.15) + $700,000 / (1.15)2
=
$1,398,865.78
Cumulative Discounted Cash Flows Year 3 = $1,000,000 / (1.15) + $700,000 / (1.15)2 +
$900,000 / (1.15)3
= $1,990,630.39
The cumulative discounted cash flows exceed the initial investment of $1,800,000 by the
end of year 3. Many companies analyze the payback period in whole years. The payback
period for the project is three years.
The discounted payback period for the submarine ride is three years.
According to the payback period rule, the projects are equally desirable.
b. Deepwater Fishing IRR:
Set the project’s cash flows, discounted at the internal rate of return (IRR), equal to zero.
Solve for the IRR.
IRR
0
IRR = C0 + C1 / (1+IRR) + C2 / (1+IRR)2 + C3 / (1+IRR)3
= $600,000+ $270,000 / (1+IRR) + $350,000 / (1+IRR)2 + $300,000 / (1+IRR)3
= 0.243 The IRR of the deepwater fishing project is 24.3%.
Submarine Ride IRR:
Set the project’s cash flows, discounted at the internal rate of return (IRR), equal to zero.
Solve for the IRR.
IRR
0
IRR = C0 + C1 / (1+IRR) + C2 / (1+IRR)2 + C3 / (1+IRR)3
= $1,800,000 + $1,000,000 / (1+IRR) + $700,000 / (1+IRR)2 +
$900,000 / (1+IRR)3
= 0.2146 The IRR of the submarine ride is 21.46%.
Based on the IRR rule, the deepwater fishing project should be chosen because it
has the higher IRR.
c. Calculate the IRR of the incremental cash flows, defined as the New Saturday Late
Edition’s Cash Flows minus the New Sunday Early Edition’s cash flows. Then calculate
the IRR of the incremental cash flows. Remember to subtract the cash flows of the
B105 project with the smaller initial investment from those of the project with the larger initial
investment, so that the incremental initial investment is negative.
Submarine Ride
Deepwater Fishing
Submarine  Fishing
IRR
0
IRR Year 0
1,800,000
600,000
$1,200,000 Year 1
1,000,000
270,000
$730,000 Year 2
700,000
350,000
$350,000 Year 3
900,000
300,000
$600,000 = C0 + C1 / (1+IRR) + C2 / (1+IRR)2 + C3 / (1+IRR)3
= $1,200,000 + $730,000 / (1+IRR) + $350,000 / (1+IRR)2 +
$600,000 / (1+IRR)3
= 0.1992 For investingtype projects, accept the larger project when the incremental IRR is
greater than the discount rate. Since the incremental IRR, 19.92%, is greater than
the required rate of return, 15%, choose the submarine ride project. Note that the
choice in part (c) differs from the choice in part (b). The choice in part (b) is flawed
because there is a scale problem. That is, the submarine ride has a greater initial
investment than does the deepwater fishing project. This problem is corrected only
by calculating the IRR of the incremental cash flows.
d. NPV(Deepwater Fishing) = $600,000 + $270,000 / (1.15) + $350,000 / (1.15)2 +
$300,000 / (1.15)3
= $96,687.76
The NPV of the deepwater fishing project is $96,687.76.
NPV(Submarine Ride) = $1,800,000 + $1,000,000 / (1.15) + $700,000 / (1.15)2 +
$900,000 / (1.15)3
= $190,630.39 The NPV of the submarine ride project is $190,630.39.
Since the NPV of the submarine ride project is greater than the NPV of the
deepwater fishing project, choose the submarine ride project. The NPV rule is
always consistent with the incremental IRR rule.
6.20 a. The project involves three cash flows: the initial investment, the annual cash inflows, and
the abandonment costs. The mine will generate cash inflows over its 11year economic
life. To express the PV of the annual cash inflows, apply the growing annuity formula,
discounted at the IRR and growing at eight percent.
PV(Cash Inflows) = C1 GATIRR, g *
= $100,000 GA11IRR, 0.08 * The notation GATr, g represents a growing annuity consisting of T payments growing at a
rate of g per payment, discounted at r.
At the end of 11 years, the Utah Mining Corporate will abandon the mine, incurring a
$50,000 charge. Discount that charge back 11 years at the IRR to express its PV.
PV(Abandonment) = C11 / (1+IRR)11
= $50,000 / (1+IRR)11 Solve the expression that equates the cash flows, discounted at the IRR, with zero. Solve
for the IRR. Use a graphing calculator.
B106 IRR
0
0
IRR = Initial Investment + PV(Cash Inflows) – PV(Abandonment)
= C0 + C1 GATIRR, g  C11 / (1+IRR)T
= $600,000 + $100,000 GA11IRR, 0.08  $50,000 / (1+IRR)11
= 0.185565 The IRR of the mining project is approximately 18.56%.
b. Yes. Since the mine’s IRR exceeds the required return of 10%, the mine should be
opened. The correct decision rule for an investmenttype project is to accept the project
if the discount rate is above the IRR. Anytime there is a second change in sign, there is a
possibility for multiple IRRs. By using a graphing calculator, one can determine that in
this problem, there is only one IRR. Therefore, a decision can be made. B107 621 a) Worst Case
Period
Cash Flow
PV 12% (CF) 0
$ (250,000)
= CF
(250,000) 1
0
= CF/1.12
0 2
0
= CF/(1.12)2
0 3
0
= CF/(1.12)3
0 4
0
= CF/(1.12)4
0 5
250,000
= CF/(1.12)5
141,857 NPV ($180,143) The worst case NPV is –$108,143 and will occur when the project has no cash flows until year five, when it returns $250,000.
b) The best case NPV can be an infinite amount. This is because payback period only tells you when you recover your investment. However,
the cash flows are not limited to five years and could go on forever.
6.22 a) Payback period for Project A is at year 3.25 and for Project B is at year 1.5.
Therefore, Totally Electric should select Project B because of the shorter payback period. b) B108 Avg Net Income
Avg Investment
AAR
Project A
$37,500
$75,000
.50000
Project B
$13,250
$40,000
.33125
Totally Electric should select Project A because it has the highest average accounting return.
c)
Project A Period
Cash Flow
PV 10% (CF) Project B Cash Flow 0
$(150,000)
= CF
(150,000)
$ (80,000) 1
15,000
= CF/1.1
13,636
60,000 2
35,000
= CF/(1.1)2
28,926
40,000 3
50,000
= CF/(1.1)3
37,566
20,000 4
200,000
= CF/(1.1)4
136,603
13,000 = CF
= CF/1.1
= CF/(1.1)2
= CF/(1.1)3
= CF/(1.1)4
(80,000)
54,545
33,058
15,026
8,879
NPV of Project A is $66,730 and of Project B is $31,509, so Totally Electric should select Project A because it has a higher NPV. NPV $66,730 PV 10% (CF) $31,509 d)
Project A PV IRR
= CF
= CF/IRR
= CF/(IRR)2 = CF/(IRR)3 = CF/(IRR)4
(CF)
(150,000)
12,206
23,174
26,939
87,681
$0
Project B
PV IRR
= CF
= CF/IRR
= CF/(IRR)2 = CF/(IRR)3 = CF/(IRR)4
(CF)
(80,000)
44,983
22,483
8,428
4,107
$0
IRR of Project A is 22.9% and of Project B is 33.4%, so Totally Electric should select Project B because it has a higher IRR. IRR 22.9% IRR 33.4% 6.23
Period
Cash Flow Plasma 1
1,000 2
600 3
300 PV 10% (CF) HDTV 0
$(1,000)
= CF
(1,000)
$(2,200)
= CF
(2,200) = CF/1.1
909
700
= CF/1.1
636 = CF/(1.1)2
496
1,100
= CF/(1.1)2
909 = CF/(1.1)3
225
2,200
= CF/(1.1)3
1,653 Cash Flow
PV 10% (CF) Sum of
Cash Flow Profitability
Index NPV $1,630 1.63 $630 $3,198 1.45 $998 B109 a) HDTV has a PI of 1.63 and Plasma has a PI of 1.45, therefore, Pinnacle should invest in HDTV technology.
b) HDTV has a NPV of $630 and Plasma has a NPV of $998, therefore, Pinnacle should invest in Plasma technology.
c)
Incremental
Period
0
1
2
3
Cash Flow
PI
NPV
PlasmaHDTV
Cash Flow
$(1,200)
$(300)
$500
$1,900
PV 10% (CF)
= CF
= CF/1.1
= CF/(1.1)2
= CF/(1.1)3
(1,200)
(273)
413
1,427
$1,568
1.31
$368
The profitability index, like the IRR, is a ratio and therefore limited by the problem of scale.
Because these investments are mutually exclusive, the Plasma investment should be selected because it has a higher NPV and the incremental PI is
greater than 1.
6.24 a) Payback period for Board game is in year 1 and for CDRom is at year 1.5, so Mario Brothers should select Board game. b)
0
1
2
3
$(300)
400
100
100
= CF
= CF/1.1
= CF/(1.1)2
= CF/(1.1)3
(300)
364
83
75
CDRom
Cash Flow
$(1,500)
1,100
800
400
PV 10% (CF)
= CF
= CF/1.1
= CF/(1.1)2
= CF/(1.1)3
(1,500)
1,000
661
301
From the analysis, NPV of Board game is $221 and of CDRom is $462, so Mario Brothers should select CDRom.
Board game Period
Cash Flow
PV 10% (CF) NPV $221 $462 c)
Board game = CF
= CF/IRR
= CF/(IRR)2
= CF/(IRR)3
(300)
242
36
22
$0
2
3
CDRom
PV IRR (CF)
= CF
= CF/IRR
= CF/(IRR)
= CF/(IRR)
(1,500)
846
473
182
$0
From the analysis, IRR of Board game is 65.6% and of CDRom is 30.1%, so Mario Brothers should select Board game.
PV IRR (CF) IRR 65.6% IRR 30.1% d)
CDRom Cash flow
$(1,200)
$700
$700
$300
PV IRR (CF)
(1,200)
571
466
163
Because the incremental IRR is 22.6% (>20% required IRR), the CDRom project should be selected. 22.6%
$0 IRR Sum of
Cash Flow Profitability
Index NPV $39 3.89 $29 $90 4.48 $70 $126 4.21 $96 6.25 CDMA B110 Period
Cash Flow
PV 10% (CF) G4 Cash Flow
PV 10% (CF) WiFi Cash Flow
PV 10% (CF) a) 0
$(10)
= CF
(10)
$(20)
= CF
(20)
$(30)
= CF
(30) 1
25
= CF/1.1
23
20
= CF/1.1
18
20
= CF/1.1
18 2
15
= CF/(1.1)2
12
50
= CF/(1.1)2
41
40
= CF/(1.1)2
33 3
5
= CF/(1.1)3
4
40
= CF/(1.1)3
30
100
= CF/(1.1)3
75 G4 has a PI of 4.48, WiFi has a PI of 4.21 and CDMA has a PI of 3.89. b) WiFi has a NPV of $96, G4 has a NPV of $70 and CDMA has a NPV of $29. c)
Combined
0
1
2
3
Cash Flow
PI
NPV
$(30)
$45
$65
$45
= CF
= CF/1.1
= CF/(1.1)2
= CF/(1.1)3
(30)
41
54
34
$128
4.28
$98
Although the WiFi investment has the single largest NPV, this is not the best investment for Hanmi group.
Because these investments are independent, and not mutually exclusive, Hanmi can invest in both CDMA ($10 mil) and G4 ($20 mil) which has a
combined NPV of $98 (>$96 for WiFi) and still be within the $30 million investment budget.
CDMA + G4 6.26 a) Period
Cash Flow
PV 10% (CF) Payback period for AZM is at year 1 and for AZF is at year 2, so project AZM should be taken. b)
0
1
2
($200,000)
200,000
150,000
= CF
= CF/1.1
= CF/(1.1)2
($200,000)
181,818
123,967
AZF
Cash Flow
($500,000)
200,000
300,000
PV 10% (CF)
= CF
= CF/1.1
= CF/(1.1)2
($500,000)
181,818
247,934
From the analysis, NPV of AZM is $218,482 and of AZF is $155,147, so project AZM should be taken.
AZM B111 c)
AZM Period
Cash Flow
PV 10% (CF) = CF
= CF/IRR
= CF/(RR)2
= CF/(IRR)3
($200,000)
117,616
51,876
30,507
AZF
PV IRR (CF)
= CF
= CF/IRR
= CF/(IRR)2
= CF/(IRR)3
($500,000)
159,105
189,858
151,037
From the analysis, IRR of AZM is 70% and of AZF is 26%, so project AZM should be taken. 3
150,000
= CF/(1.1)3
112,697
300,000
= CF/(1.1)3
225,394 SUM $218,482 $155,147 PV IRR (CF) $0 IRR 70% $0 IRR 26% d) Since both NPV and IRR favors project AZM, and AZM is a smaller investment to start out with, it is not necessary to conduct Incremental
IRR analysis. 6.27. a) Payback period for Dry Prepreg is at year 2 and for Solvent Prepreg is at year 1, so project Solvent Prepreg should be taken. b)
0
1
2
3
($1,000,000)
600,000
400,000
1,000,000
= CF
= CF/1.10
= CF/(1.10)2
= CF/(1.10)3
($1,000,0000)
545,455
330,579
751,315
Solvent
Cash Flow
($500,000)
500,000
300,000
100,000
2
Prepreg
PV 10% (CF)
= CF
= CF/1.10
= CF/(1.10)
= CF/(1.10)3
($500,000)
454,545
247,934
75,131
From the analysis, NPV of Dry Prepreg is $627,348 and of Solvent Prepreg is $277,611, so project Dry Prepreg should be taken.
Dry
Prepreg Period
Cash Flow
PV 10% (CF) B112 c)
Dry
PV IRR (CF)
= CF
= CF/IRR
= CF/(IRR)2
= CF/(IRR)3
Prepreg
($1,000,000)
429,218
204,698
366,084
$0
2
3
Solvent
PV IRR (CF)
= CF
= CF/IRR
= CF/(IRR)
= CF/(IRR)
Prepreg
($500,000)
335,122
134,768
30,109
($0)
From the analysis, IRR of Dry Prepreg is 40% and of Solvent Prepreg is 49%, so project Solvent Prepreg should be taken. SUM $627,348 $277,611 IRR 40% IRR 49% d) Since Solvent Prepreg has a higher IRR, but is relatively smaller in terms of investment and NPV, Incremental IRR analysis is required.
DrySolvent Cash Flow
($500,000)
100,000
100,000
900,000
PV IRR (CF)
($500,000)
74,377
55,319
370,304
Because the incremental IRR is 34% (>30% min IRR req.), the Dry Prepreg project should be considered. $0 IRR 34% 6.28
NP30 NX20 Incremental
NP30NX20 B113 Period
Cash Flow
PV 15% (CF)
PV IRR (CF)
PI
Cash Flow
PV 15% (CF)
PV IRR (CF)
PI
Cash Flow
PV IRR (CF) 0
($100,000)
($100,000)
–$100,000 1
40,000
34,783
31,092 2
40,000
30,246
24,168 3
40,000
26,301
18,786 4
40,000
22,870
14,603 5
40,000
19,887
11,351 SUM IRR/PI $34,086
$0 ($30,000)
($30,000)
–$30,000 20,000
17,391
11,559 23,000
17,391
7,683 26,450
17,391
5,107 30,418
17,391
3,394 34,980
17,391
2,256 $56,957
$0 ($70,000)
($70,000) 20,000
20,594 17,000
18,026 13,550
14,795 9,583
10,774 5,020
5,812 ($0) –3% Period
Cash Flow
PV 15% (CF)
PV IRR (CF)
PI
Cash Flow
PV 15% (CF)
PV IRR (CF)
PI
Cash Flow
PV IRR (CF) 0
($100,000)
($100,000)
–$100,000 1
50,000
43,478
38,085 2
50,000
37,807
29,010 3
40,000
26,301
17,678 4
30,000
17,153
10,099 5
20,000
9,944
5,128 SUM IRR/PI $34,682
$0 ($200,000)
($200,000)
–$200,000 60,000
52,174
46,316 60,000
45,369
35,754 60,000
39,451
27,600 100,000
57,175
35,509 200,000
99,435
54,821 $93,604
$0 ($100,000)
($100,000) 10,000
7,776 10,000
6,047 20,000
9,404 70,000
25,595 180,000
51,178 $0 29%
1.34 73%
2.90 6.29
A B Incremental
BA Implications
NPV
IRR
Incremental IRR
PI Project B should be taken.
Project A should be taken.
Project B should be taken (29% > the minimum acceptable IRR of 25%)
Project B should be taken. 31%
1.35 30%
1.47
29% 6.30
Period
0
1
2
SUM
IRR/PI
B
Cash Flow
($200,000)
200,000
111,000
PV 20% (CF)
($200,000)
166,667
77,083
$43,750
PV IRR (CF)
–$200,000
143,142
56,858
($0)
40%
PI
1.37
C
Cash Flow
($100,000)
100,000
100,000
PV 20% (CF)
($100,000)
83,333
69,444
$52,778
PV IRR (CF)
–$100,000
61,803
38,197
$0
62%
PI
1.74
Incremental
Cash Flow
($100,000)
100,000
11,000
BC
PV IRR (CF)
($100,000)
90,909
9,091
$0
10%
Clearly, project A will not have a positive NPV and therefore is not a viable project. From the analysis, projects B and C will have the same payback
period. However, project C will have a higher NPV and IRR, therefore it should be recommended.
Use 20% discount rate for projects B and C since their risk are greater than the typical core projects of 10%. B114 6.31 Using Incremental Cash Flows: Year
0
1
2
3 Cash Flows for
Project Billion –
Project Million
–Io + 1,500
300
300
500 PV @ 12% of Cash Flow subsequent to Initial investment = 862.9
For Project Billion to be more attractive, 862.9/(–I o + 1,500) > 1 (Profitability Index should be
greater than 1)
Solving we get 637.1 < Io < 1500
6.32 a) Project A would have a higher IRR since initial investment for Project A is less than that of
Project B, if the cash flows for the two projects are identical. b) False.
c) Yes since both the cash flows as well as the initial investment are twice that of Project B. B115 ...
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