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12
Capital Budgeting
OVERVIEW
This chapter concentrates on the long-term, strategic considerations and
focuses primarily on the firm’s investment opportunities. The discussions in
the preceding chapters dealt almost entirely with per period profit maximization. That analysis was fundamentally static in nature. By contrast,
investment is fundamentally dynamic since it involves streams of expenditures and revenues over time. An essential element of any investment
decision is the proper evaluation of alternative investment opportunities
involving alternative initial outlays, expected net returns, and time horizons.
Capital budgeting is the application of the principle of profit maximization to multi-period projects. Capital budgeting involves investment decisions in which expenditures and receipts continue over a significant period
of time. In general, capital budgeting projects may be classified into one of
several major categories, including capital expansion, replacement, new
product lines, mandated investments, and miscellaneous investments. Since
every investment opportunity involves expenditures (cash outflows) and
revenues (cash inflows) that are spread out over a number of time periods,
capital budgeting is an especially critical element of effective management
decision making. Capital budgeting techniques are used to evaluate the
potential profitability of possible new product lines, to plan for the replacement of damaged or worn out (depreciated) plant and equipment, to
expand existing production capacity, to engage in research and development, to institute or expand existing worker and management training programs, or evaluate the effectiveness of a major advertising campaign.
Managerial Economics: Theory and Practice
191
Copyright © 2003 by Academic Press.
All rights of reproduction in any form reserved.
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Capital Budgeting
Capital budgeting involves the subtraction of cash outflows from cash
inflows with adjustments for differences in their values over time. Differences in the values of the flows are based on the time value of money, which
says that a dollar today is worth more than a dollar tomorrow.
There are five standard methods used to evaluate the value of alternative investment projects including payback period, discounted payback
period, net present value (NPV), internal rate of return (IRR), and modified internal rate of return (MIRR). The payback period is the number of
periods required to recover an original investment. In general, risk averse
managers prefer investments with shorter payback periods.
The net present value of a project is calculated by subtracting the discounted present value of all outflows from the discounted present value of
all inflows. The discount rate is the interest rate used to evaluate the project,
and is sometimes referred to as the cost of capital, hurdle rate, cut off rate
and required rate of return. If the net present value of an investment is
positive (negative), then the project is accepted (rejected). If the net present
value of an investment is zero, the manager is indifferent to the project.
The internal rate of return is the interest rate that equates the present
values of inflows to the present value of outflows, i.e., the rate that causes
the net present value of the project to equal zero. If the internal rate of
return is greater than the cost of capital, the project is accepted.
There are a number of problems associated with using the IRR method
for evaluating capital investment projects. One problem is the possibility of
multiple internal rates of return. Multiple internal rates of return occur
when a project that has two or more internal rates of return.
For independent projects both the NPV and the IRR methods will yield
the same accept/reject decision rules. For mutually exclusive capital investment projects the NPV and the IRR methods could result in conflicting
accept/reject decision rules. This is because the NPV method implicitly
assumes that net cash inflows are reinvested at the cost of capital, whereas
the IRR method assumes that net cash inflows are reinvested at the internal rate of return.
The modified internal rate of return (MIRR) method for evaluating
capital investment projects is similar to the IRR method in that it generates accept/reject decision rules based upon interest rate comparisons. But
unlike the IRR method, the MIRR method assumes that cash flows are
reinvested at the cost of capital, and avoids some of the problems associated with multiple internal rates of return.
There are several categories of cost of capital, including the cost of debt,
cost of equity, and the weighted cost of capital. The cost of debt is the interest rate that must be paid on after-tax debt. The weighed cost of capital is
a measure of the overall cost of capital. It is obtained by weighting the
various costs by the relative proportion of each component’s value in the
total capital structure.
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Multiple Choice Questions
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MULTIPLE CHOICE QUESTIONS
12.1
The process of selecting from alternative long-term investment
projects is called:
A. Net cash inflow maximization.
B. Capital budgeting.
C. Discounting cash inflows.
D. Cash flow management.
E. Capital rationing.
12.2
The time value of money refers to:
A. The earning power of an investment or stream of investments
over time.
B. The opportunity cost of capital.
C. The interest rate earned on an investment.
D. The discount rate used to calculate the present value of an
investment.
12.3
The future value of a lump sum payment is worth $10,000 at the
end of 6 years. Suppose that the interest rate is 8 percent
compounded semiannually.
I. The present value of the $10,000 is greater if the interest rate
is compounded monthly instead of semiannually.
II. The effective annual rate of return is greater an 8 percent.
III. The semiannual interest rate is 4 percent.
Which of the following is correct?
A. I only.
B. II only.
C. III only.
D. I and II only.
E. II and III are correct.
12.4
Future value may be defined as:
A. The discounted value of future cash flows.
B. The interest rate earned on future cash flows.
C. The compounded value of future cash flows.
D. The opportunity costs of future cash flows.
E. The per period maximization of future cash flows.
12.5
Present value may be defined as:
A. The discounted value of future cash flows.
B. The interest rate earned on future cash flows.
C. The compounded value of future cash flows.
D. The opportunity costs of future cash flows.
E. The per period maximization of future cash flows.
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Capital Budgeting
12.6
If the interest (discount) rate is positive, then:
A. The present value of a series of cash flows will be greater than
its future value.
B. The future value of a series of cash flows will be greater than
its present value.
C. The present value of a series of cash flows will equal its future
value.
D. The present value is only greater than the future value for an
annuity.
12.7
Suppose that Zoe invests $25,000 into a certificate of deposit that
pays 7 percent, compounded annually. How much will Zoe’s
certificate of deposit be worth in 10 years?
A. $47,708.42.
B. $49,178.78.
C. $50,039.93.
D. $50,241.53.
E. $50,343.82.
12.8
Suppose the Zoe invests $25,000 into a certificate of deposit that
pays 7 percent, compounded quarterly. How much will Zoe’s
certificate of deposit be worth in 10 years?
A. $47,708.42.
B. $49,178.78.
C. $50,039.93.
D. $50,241.53.
E. $50,343.82.
12.9
Suppose the Zoe invests $25,000 into a certificate of deposit that
pays 7 percent, compounded monthly. How much will Zoe’s
certificate of deposit be worth in 10 years?
A. $47,708.42.
B. $49,178.78.
C. $50,039.93.
D. $50,241.53.
E. $50,343.82.
12.10 Suppose the Zoe invests $25,000 into a certificate of deposit that
pays 7 percent, compounded continuously. How much will Zoe’s
certificate of deposit be worth in 10 years?
A. $47,708.42.
B. $49,178.78.
C. $50,039.93.
D. $50,241.53.
E. $50,343.82.
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Multiple Choice Questions
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12.11 A series of fixed payments that are made fixed intervals for a
specified period of time is called:
A. An annuity.
B. A cash flow.
C. A mutually exclusive payments.
D. A payback period.
E. Compounding.
12.12 A series of fixed payments that are made fixed intervals at the end
of each period is called:
A. An annuity due.
B. An ordinary annuity.
C. A payback annuity.
D. Discounting.
E. Compounding.
12.13 A series of fixed payments that are made fixed intervals at the
beginning of each period is called:
A. An annuity due.
B. An ordinary annuity.
C. A payback annuity.
D. Discounting.
E. Compounding.
12.14 Cletus invests $2,000 annually in an ordinary annuity that pays 9
percent interest compounded annually. The future value of this
annuity in 10 years is:
A. $28,333.33.
B. $29,672.21.
C. $30,385.86.
D. $32,111.11.
E. $33,707.03.
12.15 Cletus invests $2,000 annually in an annuity due that pays 9
percent interest compounded annually. The future value of this
annuity in 10 years is:
A. $28,333.33.
B. $29,672.21.
C. $30,385.86.
D. $33,120.59.
E. $33,707.03.
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Capital Budgeting
12.16 Cletus invests $2,000 annually in an annuity due that pays 9
percent interest compounded quarterly. The future value of this
annuity in 10 years is:
A. $28,333.33.
B. $29,672.21.
C. $30,385.86.
D. $33,120.59.
E. $33,707.03.
12.17 Suppose that Cletus wants a lump-sum investment today to grow
to $100,000 in 25 years. If Cletus can reasonably expect to earn 8.5
percent compounded annually, then the lump-sum investment
should be:
A. $11,943.30.
B. $12,033.10.
C. $12,212.18.
D. $12,479.49.
E. $13,009.38.
12.18 Suppose that Cletus wants a lump-sum investment today to grow
to $100,000 in 25 years. If Cletus can reasonably expect to earn 8.5
percent compounded semiannually, then the lump-sum investment
should be:
A. $11,943.30.
B. $12,033.10.
C. $12,212.18.
D. $12,479.49.
E. $13,009.38.
12.19 Suppose that Cletus wants a lump-sum investment today to grow
to $100,000 in 25 years. If Cletus can reasonably expect to earn 8.5
percent compounded quarterly, then the lump-sum investment
should be:
A. $11,943.30.
B. $12,033.10.
C. $12,212.18.
D. $12,479.49.
E. $13,009.38.
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Multiple Choice Questions
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12.20 Suppose that Cletus wants a lump-sum investment today to grow
to $100,000 in 25 years. If Cletus can reasonably expect to earn 8.5
percent compounded monthly, then the lump-sum investment
should be:
A. $11,943.30.
B. $12,033.10.
C. $12,212.18.
D. $12,479.49.
E. $13,009.38.
12.21 Suppose that Cletus wants a lump-sum investment today to grow
to $100,000 in 25 years. If Cletus can reasonably expect to earn 8.5
percent compounded continuously, then the lump-sum investment
should be:
A. $11,943.30.
B. $12,033.10.
C. $12,212.18.
D. $12,479.49.
E. $13,009.38.
12.22 Suppose that Cletus decides to invest $4,000 per year into a 25 year
annuity due that earns an interest rate of 8.5 percent compounded
annually. Calculate the present value of Cletus’ investment plan?
A. $40,936.76.
B. $41,286.75.
C. $42,821.33.
D. $42,977.77.
E. $44,416.39.
12.23 Suppose that Cletus decides to invest $4,000 per year into a 25 year
ordinary annuity that earns an interest rate of 8.5 percent
compounded annually. Calculate the present value of Cletus’
investment plan?
A. $40,936.76.
B. $41,286.75.
C. $42,821.33.
D. $42,977.77.
E. $44,416.39.
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12.24 Suppose that Cletus has decided to invest in a retirement annuity.
Cletus’ goal is to have $1,000,000 in his annuity by the time that he
is 65 years old. Cletus is confident of earning a 6 percent interest
rate compounded annually. Cletus is currently 30 years old and
plans to make his first investment today. How much will Cletus
have to invest annually to reach is goal?
A. $8,125.32.
B. $8,465.90.
C. $8,973.86.
D. $9,026.13.
E. None of the above.
12.25 Suppose that Chloe borrows $300,000 from the First National State
Bank at 2.5 percent interest compounded annually to purchase a
new home. Chloe agrees to repay the loan in 30 equal annual
installments, with the first payment due at the end of the first year.
How much are Chloe’s annual payments?
A. $14,333.25.
B. $15,666.35.
C. $16,777.45.
D. $17,888.55.
E. None of the above.
12.26 The payback period is:
A. The number cash-flow periods of an capital investment project.
B. The number of years that it takes to earn a profit of a capital
investment project.
C. The number of periods required to pay for the original
investment.
D. The number of periods required to calculate the net present
value of an investment project.
12.27 An advantage of the payback period method of evaluating a
capital investment project is that it:
A. Does not consider the time value of money.
B. Ignores cash flows beyond the payback period.
C. Provides a rough approximation of a projects liquidity and risk.
D. Provides a rough approximation of the present value of net
cash flows.
E. None of the above.
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Multiple Choice Questions
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12.28 An advantage of the discounted payback period method of
evaluating a capital investment project is that it:
A. Considers the time value of money.
B. Ignores cash flows beyond the payback period.
C. Provides a rough approximation of a projects liquidity and risk.
D. Provides a rough approximation of the present value of net
cash flows.
E. Both A and C are correct.
12.29 Suppose that the payback period for a particular project is 5 years
and 6 months. If the annual cash inflows are $5,000, then the initial
investment was:
A. $22,500.
B. $24,000.
C. $27,500.
D. $29,000.
E. None of the above.
12.30 Two project are independent if:
A. Acceptance of one project means rejection of the other.
B. Their cash flows are unrelated.
C. They have different hurdle rates.
D. They have different discounted payback periods.
12.31 The cost of capital is:
A. The cost of acquiring funds to finance a capital investment
project.
B. The minimum rate of return that must be earned to justify a
capital investment.
C. The same thing as the required rate of return.
D. Also referred to as the hurdle rate.
E. All of the above statements are true.
12.32 Suppose that a project with an initial investment of $50,000 is
expected to generate an annual cash flow of $4,000 for each of the
next 7 years. This project should not be accepted if the cost of
capital is:
A. 8 percent.
B. 9 percent.
C. 10 percent.
D. 11 percent.
E. Both C and D are correct.
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12.33 Suppose that a project with an initial investment of $30,000 has the
following annual cash inflows:
Year
Cash inflow
1
$4,000
2
$3,500
3
$8,000
4
5
$12,000 $8,000
If the cost of capital is 8 percent, then the net present value of the
project:
A. -$3,506.37–the project should be rejected.
B. -$4,708.42–the project should be rejected.
C. $3,506.37–the project should be accepted.
D. $4,708.42–the project should be accepted.
E. None of the above statements are true.
12.34 The internal rate of return (IRR) is:
A. The same thing as the discount rate.
B. The same thing as the cost of capital.
C. The discount rate that equates the present values of inflows and
outflows.
D. The same thing as the net present value.
E. The ratio of average annual profits to average investments.
12.35 Project A and Project B are mutually exclusive. Project A has an
IRR of 10 percent. Project B has an IRR of 12 percent. If the
marginal cost of capital is 11 percent, then:
A. Project A should be accepted and Project B rejected.
B. Project B should be accepted and Project A rejected.
C. Both projects should be accepted since the decision is not
based on the IRR but the NPV.
D. Both projects should be rejected since the decision is not based
on the IRR but the NPV.
12.36 Project A and Project B are independent. Project A has an IRR of
12 percent. Project B has an IRR of 14 percent. If the marginal cost
of capital is 10 percent, then:
A. Project A must have a higher NPV than Project B.
B. Project B must have a higher NPV than Project A.
C. The NPV of both projects must be negative.
D. The NPV of both projects is positive.
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Multiple Choice Questions
12.37 Suppose that a firm is considering several mutually exclusive
projects. The firm should choose:
A. The project with the highest NPV.
B. The project with the lowest NPV.
C. All projects with a positive NPV.
D. The project with the lowest IRR.
E. The project with the lowest cost of capital.
12.38 Suppose that a project with an initial investment of $30,000 has the
following annual cash inflows:
Year
Cash inflow
1
$4,000
2
$3,500
3
$8,000
4
$12,000
5
$8,000
Using a financial calculator, the IRR for this project is:
A. 3.75 percent.
B. 4.17 percent.
C. 4.71 percent.
D. 5.04 percent.
E. None of the above.
12.39 Suppose that a firm is considering to independent projects. The
crossover rate is:
A. The IRR at which the NPV of the two projects are equal.
B. The cost of capital at which the NPV of the two projects are
equal.
C. Is the discount rate at which the NPV of the two projects are
equal.
D. Is the discount rate at which the discounted payback period of
the two projects are equal.
TABLE 1
Net Cash Flows for Projects Amber and Jade
Year (t)
Project Amber
Project Jade
0
1
2
3
-$3,000
1,000
1,000
2,500
-$4,000
1,750
1,500
2,000
12.40 Consider the information presented in Table 1. If the discount rate
is 9 percent, the NPV of Project Amber is:
A. $412.39.
B. $598.78.
C. $689.57.
D. $724.44.
E. None of the above.
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12.41 Consider the information presented in Table 1. If the discount rate
is 9 percent, the NPV of Project Jade is:
A. $412.39.
B. $598.78.
C. $689.57.
D. $724.44.
E. None of the above.
12.42 Consider the information presented in Table 1. Suppose that
Projects Amber and Jade are mutually exclusive. If the discount
rate is 9 percent, then:
A. Project Jade is preferred to Project Amber.
B. Project Amber is preferred to Project Jade.
C. Project Amber is equivalent to Project Jade.
D. Both projects will be chosen since the NPV is positive.
E. Neither projects will be chosen since they are mutually
exclusive.
12.43 Consider the information in Table 1. The IRR for Project Amber is:
A. 10.17 percent.
B. 12.47 percent.
C. 14.57 percent.
D. 19.54 percent.
E. None of the above.
12.44 Consider the information in Table 1. The IRR for Project Jade is:
A. 10.17 percent.
B. 12.47 percent.
C. 14.57 percent.
D. 19.54 percent.
E. None of the above.
12.45 When the cost of capital is less than IRR for two mutually
exclusive projects, then:
A. The NPV and IRR methods will always result in the same
accept and reject decisions.
B. The NPV method will lead to an accept decision while the IRR
method will lead to a reject decision.
C. The IRR method will lead to an accept decision while the NPV
method will lead to a reject decision.
D. The project with the highest IRR should be chosen.
E. Both A and E are correct.
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Multiple Choice Questions
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12.46 Cyborg Electronics is considering two mutually exclusive capital
investment projects. Project A has an IRR of 10 percent. Project B
has an IRR of 12 percent. The crossover rate is 8 percent. Cyborg
should:
I. Invest in both projects if the cost of capital is less than 10
percent.
II. Invest in Project A if the cost of capital is less than 8 percent.
III. Invest in Project B if the cost of capital is greater than 8
percent but less than 12 percent.
IV. Invest in Project A if the cost of capital is greater than 8
percent but less than 10 percent.
Which of the following is correct?
A. I only.
B. II only.
C. III only.
D. II and III only.
E. II and IV only.
12.47 Cyborg Electronics is considering two independent capital
investment projects. Project A has an IRR of 10 percent. Project B
has an IRR of 12 percent. The crossover rate is 8 percent. Cyborg
should:
I. Invest in both projects if the cost of capital is less than 10
percent.
II. Invest in Project A if the cost of capital is less than 8 percent.
III. Invest in Project B if the cost of capital is greater than 8
percent but less than12 percent.
IV. Invest in Project A if the cost of capital is greater than 8
percent but less than 10 percent.
Which of the following is correct?
A. I only.
B. II only.
C. III only.
D. I and III only.
E. I, II, III and IV.
12.48 Which of the following statements about the modified internal rate
of return (MIRR) is correct?
A. The assumption regarding reinvestment underlying the MIRR
method is more reasonable than that underlying the IRR
method.
B. The selection of a capital investment project using the MIRR
method is always consistent with that of the IRR method.
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C. The MIRR method always overcomes the problems associated
with multiple IRR.
D. All of the above are correct.
E. All of the above are incorrect.
12.49 The cost of debt capital is:
A. The interest rate that must be paid on the debt.
B. The after-tax interest rate that must be paid on the debt.
C. The future value of the debt less principal.
D. The equivalent rate of return on the company’s equity.
E. The required rate of return on a company’s stock.
12.50 The optimal capital structure of a firm:
A. Minimizes the firm’s cost of debt capital.
B. Is that combination of debt, preferred and common stock that
maximizes the firm’s share values.
C. Is one in which the weighted cost of capital is less than the
IRR.
D. All of the above are correct.
E. None of the above are correct.
SHORTER PROBLEMS
12.1
Adam wants to borrow $15,000 at 7.5 percent interest compounded
monthly from the Cedar Federal Credit Union to purchase a new
car. If the loan principal is to be repaid in a lump sum, how much
will Adam repay the bank in 3 years?
12.2
Andrew deposits $25,000 in a certificate of deposit that pays 6
percent interest compounded continuously. How much money will
Andrew’s certificate of deposit be worth at the end of 25 years?
12.3
Alex wants to invest $2,500 a year into an annuity that can
reasonably expect to earn 7 percent compounded annually. If Alex
makes his first $2,500 investment immediately, how much will his
annuity be worth in 20 years?
12.4
Alex wants to invest $2,500 a year into an annuity that can
reasonably expect to earn 7 percent compounded annually.
Suppose that Alex is uncertain as to whether to open the annuity
immediately, or wait until the end of the year to make his first
deposit since he can earn $250 in interest the first year by lending
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Longer Problems
$2,500 to his brother, Adam. Suppose that Alex plans to deposit
the $2,750 into a certificate of deposit earning 6 percent interest
compounded annually for 19 years. Should Alex loan $2,500 to
Adam, or should he open the annuity immediately?
12.5
Suppose that Folderol Savings and Loan is offering 10-year
certificates of deposit earning a 7.5 interest rate compounded
quarterly. How much would Nina have to deposit today in order
for the certificate of deposit to be worth $250,000 at the end of 10
years?
12.6
Suppose that Zelda borrows $500,000 at 6.5 percent to purchase a
luxury condominium in downtown Toledo, Ohio. Zelda will to
repay the loan in 20 equal annual installments, with the first
payment due at the end of the first year.
A. What are Zelda’s annual mortgage payments?
B. How much interest will Zelda be paying?
12.7
Calculate the weighted average cost of capital of a project that is
35 percent debt and 65 percent equity. Assume that the firm pays
10 percent on debt and 15 percent on equity. Assume that the
firm’s marginal tax rate is 33 percent.
LONGER PROBLEMS
12.1
The Finance & Economics Department of the Hobgoblin School of
Business is considering purchasing a new photocopying machine
for use by faculty and doctoral candidates. Dr. Windsock, the
department chair, has asked Dr. Steadfast, a finance professor, to
determine which of two models, the Galaxy 5000 or the Nova 2700,
should be purchased. In addition to the cost of the photocopy
machine itself, the manufacturer of each model offers 5-year
service contract. The cash outflows for each photocopy machine is
summarized in the following table.
Net Cash Flows (CFt) for the Galaxy 5000 and Nova 2700
Year (t)
Galaxy 5000
Nova 2700
0
1
2
3
4
5
-$10,500
-320
-320
-320
-320
-320
-$11,500
-230
-230
-230
-230
-230
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If the cost of capital is 5.5 percent compounded annually, which
photocopy machine should the department purchase if the
estimated salvage values for the Galaxy 500 and Nova 2700 after
five years are $1,000 and $2,000, respectively?
12.2
Samuel Adams of Niagra Company is a leading producer of natural
gas. Adams is considering two mutually exclusive projects involving
drilling operations Alaska. The projected net cash flows for each
project are summarized in the following table.
Net Cash Flows (CFt) for Projects A and B ($millions)
Year (t)
Project A
Project B
0
1
2
3
-$4,000
2,000
3,000
-$5,000
2,000
2,500
3,000
Determine which project should be accepted if the cost of capital is
9 percent.
12.3
Mandalay Enterprises is considering two mutually exclusive
projects. The projected net cash flows for Projects A and B are
summarized in the following table.
Net Cash Flows (CFt) for Projects A and B
Year (t)
Project A
Project B
0
1
2
3
4
5
-$27,000
8,000
9,000
10,000
10,000
6,000
-$20,000
6,500
6,500
6,500
6,500
6,500
A. Calculate the NPV for both projects if the cost of capital is 15
percent.
B. Based on your answer to part A, which project should be
accepted?
C. Calculate the IRR for both projects.
D. Based on your answer to part C, which project should be
accepted?
E. If Projects A and B are independent and the cost of capital is
18.5 percent, then which project(s) should be accepted?
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Answers to Multiple Choice Questions
12.4
Consider, again, the net cash flows the two projects being
considered by Mandalay Enterprises in the previous question.
A. Illustrate the net present value profiles for Projects A and B.
B. What is the crossover rate for the two projects?
C. Assuming that Projects A and B are mutually exclusive, which
project should be selected if the cost of capital is greater than
the crossover rate? Which project should be selected if the cost
of capital is less than the crossover rate?
12.5
Consider the following cash flows for a capital investment project:
Net Cash Flows (CFt)
Year (t)
CFt
0
1
2
-$1,000
5,000
-6,000
A. Summarize the projects net present value profile for selected
costs of capital?
B. Does the project have multiple internal rates of return? What
are they?
C. What is the IRR that maximizes the projects NPV?
D. Diagram your answer.
ANSWERS TO MULTIPLE CHOICE QUESTIONS
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
12.10
12.11
12.12
12.13
B.
A.
D.
C.
A.
B.
B.
C.
D.
E.
A.
B.
A.
12.14
12.15
12.16
12.17
12.18
12.19
12.20
12.21
12.22
12.23
12.24
12.25
12.26
C.
D.
E.
E.
D.
C.
B.
A.
E.
A.
B.
A.
C.
12.27
12.28
12.29
12.30
12.31
12.32
12.33
12.34
12.35
12.36
12.37
12.38
12.39
C.
E.
C.
B.
E.
E.
A.
C.
B.
D.
A.
D.
B.
12.40
12.41
12.42
12.43
12.44
12.45
12.46
12.47
12.48
12.49
12.50
C.
A.
B.
D.
C.
A.
D.
E.
A.
B.
B.
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Capital Budgeting
SOLUTIONS TO SHORTER PROBLEMS
12.1
FVn = PV0(1 + i/m)mn
FV3 = $15,000(1 + 0.075/12)3¥12
= $15,000(1.00625)36
= $15,000(1.25145)
= $18,771.69
12.2
FVn = PV0ein
FV25 = $25,000e0.06¥25
= $25,000e1.5
= $25,000(4.4817)
= $112,042.23
12.3
This is an example of an annuity due. The future value of Alex’s
annuity will be
FVADn = A{[(1 + i)n - 1]/i}(1 + i)
= $2,500{[(1.07)20 - 1]/0.07}(1.07)
= $2,500[(3.8697 - 1)/0.07](1.07)
= $2,500(40.995)(1.07)
= $109,663
12.4
From the previous problem we saw that if Alex opens the annuity
immediately, then its value in 20 years will be $109,663. If Alex
waits for a year before making the first deposit then the future
value of an ordinary annuity is
FVOAn = A[(1 + i)n - 1]/i
= $2,500[(1.0720 - 1)/0.07]
= $2,500(3.8697 - 1)/0.07
= $2,500(40.9955)
= $102,488.75
To this amount must be added the future value of $2,750
compounded annually for 19 years at an interest rate of 6 percent.
This amount is given as
FVn = PV0(1 + i)n
= $2,500(1.06)19
= $2,500(3.0256)
= $7,564
Adding this amount to the future value of an ordinary annuity
yields
$102,488.75 + $7,564 = $110,052.75
which is greater than the amount that Alex can earn by opening
the annuity immediately. Thus, Alex should lend to his brother.
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Solutions to Longer Problems
209
12.5
PV0 = FVn/(1 + i/m)nm
= $250,000/(1 + 0.075/4)10¥4
= $250,000/2.10235
= $118,914.59
12.6
A. A = PVOAn/St=1Æn[1/(1 + i)]t
= $500,000/St=1Æ20[1/(1.065)]t
= $500,000/11.0185
= $45,378.23
B. Zelda will make total mortgage payments of 20(45,378.23) =
$907,564.55. Thus, the total amount of interest paid will be
$907,564.55 - $500,000 = $407,564.55.
12.7
WACC = wdkd(1 - t) + weke
= 0.35(0.1)(1 - 0.33) + 0.65(0.15)
= 0.12095 or 12.095 percent
SOLUTIONS TO LONGER PROBLEMS
12.1
The NPV for the Galaxy 5000 (NPVG) and the Nova 2700 (NPVN)
are
NPVG = CF0/(1 + k)0 + CF1/(1 + k)1 + CF2/(1 + k)2 + . . .
+ CF5/(1 + k)5
= -10,500/(1.055)0 - 320/(1.055)1 - 320/(1.055)2 - 320/(1.055)3
- 320/(1.055)4 - 320/(1.055)5 + 1,000/(1.055)5
= -$11,101.36
NPVN = -11,500/(1.055)0 - 230/(1.055)1 - 230/(1.055)2 - 230/(1.055)3
- 230/(1.055)4 - 230/(1.055)5 + 2,000/(1.055)5
= -$10,951.90
Since /NPVN/ < /NPVG/, Windsock will purchase the Nova 2700.
12.2
Since the projects have different lives, they must be compared over
the least common multiple of years, which in this case is 6 years.
NPVA = CF0/(1 + k)0 + CF1/(1 + k)1 + CF2/(1 + k)2 + . . .
+ CF6/(1 + k)6
= -4,000/(1.09)0 + 2,000/(1.09)1 + 3,000/(1.09)2 - 4,000/(1.09)2
+ 2,000/(1.09)3 + 3,000/(1.09)4 - 4,000/(1.09)4 + 2,000/(1.09)5
+ 3,000/(1.09)6
= $917.79
NPVB = -5,000/(1.09)0 + 2,000/(1.09)1 + 2,500/(1.09)2 + 3,000/(1.09)3
- 5,000/(1.09)3 + 1,000/(1.09)4 + 2,500/(1.09)5 + 3,000/(1.09)6
= $1,516.75
Since NPVB > NPVA, then Adams will choose Project B over
Project A.
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Capital Budgeting
12.3
A. NPVA = CF0/(1 + k)0 + CF1/(1 + k)1 + CF2/(1 + k)2 + . . .
+ CF5/(1 + k)5
= -27,000/(1.15)0 + 8,000/(1.15)1 + 9,000/(1.15)2
+ 10,000/(1.15)3 + 10,000/(1.15)4 + 6,000/(1.15)5
= $2,037.57
NPVB = -20,000/(1.15)0 + 6,500/(1.15)1 + 6,500/(1.15)2
+ 6,500/(1.15)3 + 6,500/(1.15)4 + 6,500/(1.15)5
= $1,789.01
B. Since NPVA > NPVB, then Project A should be accepted.
C. To determine the internal rate of return for Projects A and B,
substitute the isnformation provided in the table into the
Equation (12.27) and solve for IRR.
NPVA = CF0 + CF1/(1 + IRRA)1 + CF2/(1 + IRRA)2 + . . .
+ CF5/(1 + IRRA)5
= -$27,000 + $8,000/(1 + IRRA)1 + $9,000/(1 + IRRA)2
+ $10,000/(1 + IRRA)3 + $10,000/(1 + IRRA)4
+ $6,000/(1 + IRRA)5 = 0
NPVB = -$20,000 + $6,500/(1 + IRRB)1 + $6,500/(1 + IRRB)2
+ $6,500/(1 + IRRB)3 + $6,500/(1 + IRRB)4
+ $6,500/(1 + IRRB)5 = 0
Since calculating IRRA and IRRB by trial and error is time
consuming and tedious, the solution values were obtained by
using a financial calculator. The internal rates of return for
Projects A and B are
IRRA = 18.18 percent
IRRB = 18.72 percent
D. Since IRRB > IRRA, then Project B should be accepted.
E. The internal rate of return is greater than the hurdle rate for
Project B and less than the hurdle rate for Project B. Thus,
Project B should be accepted and Project A rejected.
12.4
A. With the use of a financial calculator, the net present values for
Projects A and B for various interest rates are summarized in
the following table.
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Solutions to Longer Problems
Net Present Value Profiles for Projects A and B
Cost of Capital
Project A
Project B
0.00
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.1657
0.18
0.20
$16,000
11,383
9,358
7,496
5,780
4,195
2,729
1,371
1,003
110
-1,063
$12,500
8,937
7,380
5,953
4,640
3,431
2,315
1,283
1,003
326
-561
B. To determine the crossover rate, equate the net present value
of Project A with the net present value of Project B and solve
for the cost of capital, k.
NPVA = NPVB
-$27,000/(1 + k)0 + $8,000/(1 + k)1 + $9,000/(1 + k)2
+ $10,000/(1 + k)3 + $10,000/(1 + k)4 + $6,000/(1 + k)5
= -$20,000/(1 + k)0 + $6,500/(1 + k)1 + $6,500/(1 + k)2
+ $6,500/(1 + k)3 + $6,500/(1 + k)4 + $6,500/(1 + k)5
Bringing all of the terms in this expression to the left-hand side
of the equation we obtain
-$7,000/(1 + k)0 + $1,500/(1 + k)1 + $2,500/(1 + k)2
+ $3,500/(1 + k)3 + $3,500/(1 + k)4 - $500/(1 + k)5 = 0
The value for k in this expression may be found using the IRR
function of a financial calculator. Solving for k we obtain
Crossover rate = 16.57 percent
C. From the previous problem, the internal rates of return for the
two projects is
IRRA = 18.18 percent
IRRB = 18.72 percent
Finally, calculating the net present value of Projects A and B
using the crossover rate yields
NPVA = CF0/(1 + k)0 + CF1/(1 + k)1 + CF2/(1 + k)2 + . . .
+ CF5/(1 + k)5
= -27,000/(1.1657)0 + 8,000/(1.1657)1 + 9,000/(1.1657)2
+ 10,000/(1.1657)3 + 10,000/(1.1657)4 + 6,000/(1.1657)5
= $1,003
NPVB = -20,000/(1.1657)0 + 6,500/(1.1657)1 + 6,500/(1.1657)2
+ 6,500/(1.657)3 + 6,500/(1.1657)4 + 6,500/(1.1657)5
= $1,003
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Capital Budgeting
With this information, the net present value profiles for Project
A and Project B may be illustrated in the following diagram.
From the above diagram, if the cost of capital is greater than
16.57 percent, but less than 18.72 percent, then Project B is
preferred to Project A since NPVB > NPVA. This choice of
projects is consistent with the IRR method since IRRB > IRRA.
On the other hand, if the cost of capital is less than 16.57
percent, then Project A is preferred to Project B since NPVA >
NPVB. This result, however, is in conflict with the choice of
projects based on the IRR method.
12.5
A.
Net Present Value Profile
k
NPV
0.00
0.10
0.25
0.50
0.75
1.00
1.25
1.40
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
-$2,000.00
-1,413.22
-840.00
-333.33
-102.04
00.00
37.04
41.67
40.00
24.79
00.00
-29.59
-61.22
-93.33
-125.00
-155.71
-185.19
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Solutions to Longer Problems
213
B. NPV = -1,000 + 5,000/(1 + IRR)1 - 6,000/(1 + IRR)2 = 0
-6,000[1/(1 + IRR)]2 + 5,000[1/(1 + IRR)] - 1,000 = 0
a[1/(1 + IRR)]2 + b[1/(1 + IRR)] + c = 0
[1/(1 + IRR)]1,2 = {-b ÷[b2 - 4ac]}/2a
= {-5,000 ± ÷[(5,000)2
- 4(-6,000)(-1,000)]}/2(-6,000)
= [-5,000 ± ÷(1,000,000)]/-12,000
= (-5,000 ± 1,000)/-12,000
[1/(1 + IRR)]1 = (-5,000 + 1,000)/-12,000 = 0.3333
(1 + IRR)1 = 3.00
IRR1 = 2.00 or 200 percent
[1/(1 + IRR)]2 = (-5,000 - 1,000)/-12,000 = 0.5
(1 + IRR)2 = 2.00
IRR2 = 1.00 or 100 percent
Thus, and this was illustrated NPV profile, the project has
internal rates of return of 100 percent and 200 percent.
C. NPV = -1,000 + 5,000(1 + IRR)-1 - 6,000/(1 + IRR)-2
dNPV/d(1 + IRR) = -5,000(1 + IRR)-2 + 12,000/(1 + IRR)-3 = 0
5,000(1 + IRR) = 12,000
1 + IRR = 12,000/5,000
IRR* = 1.40 or 140 percent
The value of IRR that maximizes the NPV of the firm is 140
percent, which is highlighted in the NPV profile.
D.
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