Power Efficiency Guide
What is the net present value of an investment, and how do you calculate it?
The net present value of a project measures the difference between its value and cost. NPV is therefore the amount that the project will add to shareholder wealth. A company maximizes shareholder wealth by accepting all projects that have a positive NPV.
How is the internal rate of return of a project calculated and what must one look out for when using the internal rate of return rule?
Instead of asking whether a project has a positive NPV, many businesses prefer to ask whether it offers a higher return than shareholders could expect to get by investing in the capital market. Return is usually defined as the discount rate that would result in a zero NPV. This is known as the internal rate of return, or IRR. The project is attractive if the IRR exceeds the opportunity cost of capital.
There are some pitfalls in using the internal rate of return rule. Be careful about using the IRR when (1) the early cash flows are positive, (1) there is more than one change in the sign of the cash flows, or (3) you need to choose between two mutually exclusive projects.
Why don't the payback rule and book rate of return rule always make shareholders better off?
10 Unfortunately, when capital is rationed in more than one period, or when personnel, production capacity, or other resources are rationed in addition to capital, it isn't always possible to get the NPV-maximizing package just by ranking projects on their profitability index. Tedious trial and error may be called for, or linear programming methods may be used.
Brealey-Myers: I II. Value I 6. Net Present Value and
Fundamentals of Corporate Other Investment Criteria
Finance, Third Edition
Brealey-Myers: Fundamentals of Corporate Finance, Third Edition
II. Value
6. Net Present Value and Other Investment Criteria
© The McGraw-Hill Companies, 2001
The net present value rule and the rate of return rule both properly reflect the time value of money. But companies sometimes use rules of thumb to judge projects. One is the payback rule, which states that a project is acceptable if you get your money back within a specified period. The payback rule takes no account of any cash flows that arrive after the payback period and fails to discount cash flows within the payback period.
Book (or accounting) rate of return is the income of a project divided by the book value. Unlike the internal rate of return, book rate of return does not depend just on the project's cash flows. It also depends on which cash flows are classified as capital investments and which as operating expenses. Managers often keep an eye on how projects would affect book return.
How can the net present value rule be used to analyze three common problems that involve competing projects: when to postpone an investment expenditure; how to choose between projects with equal lives; and when to replace equipment?
Sometimes a project may have a positive NPV if undertaken today but an even higher NPV if the investment is delayed. Choose between these alternatives by comparing their NPVs today.
When you have to choose between projects with different lives, you should put them on an equal footing by comparing the equivalent annual cost or benefit of the two projects. When you are considering whether to replace an aging machine with a new one, you should compare the cost of operating the old one with the equivalent annual cost of the new one.
How is the profitability index calculated, and how can it be used to choose between projects when funds are limited?
If there is a shortage of capital, companies need to choose projects that offer the highest net present value per dollar of investment. This measure is known as the profitability index.
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Key Terms opportunity cost of capital net present value (NPV) internal rate of return (IRR) payback period book rate of return capital rationing profitability index
(accounting rate of return) mutually exclusive projects equivalent annual cost
Quiz
Problems 1-9 refer to two projects with the following cash flows:
Year Project A Project B
J6<r chapter 6 Net Present Value and Other Investment Criteria 191
IRR/NPV. If the opportunity cost of capital is 11 percent, which of these projects is worth pursuing?
Mutually Exclusive Investments. Suppose that you can choose only one of these projects. Which would you choose? The discount rate is still 11 percent.
IRR/NPV. Which project would you choose if the opportunity cost of capital were 16 percent?
IRR. What are the internal rates of return on projects A and B?
Investment Criteria. In light of your answers to problems 2-4, is there any reason to believe that the project with the higher IRR is the better project?
Profitability Index. If the opportunity cost of capital is 11 percent, what is the profitability index for each project? Does the profitability index rank the projects correctly? Payback. What is the payback period of each project?
Investment Criteria. Considering your answers to problems 2, 3, and 7, is there any reason to believe that the project with the lower payback period is the better project? Book Rate of Return. Accountants have set up the following depreciation schedules for the two projects:
Year:
Project A Project B
Calculate book rates of return for each year. Are these book returns the same as the IRR?
10. NPV and IRR. A project that costs $3,000 to install will provide annual cash flows of $800 for each of the next 6 years. Is this project worth pursuing if the discount rate is 10 percent? How high can the discount rate be before you would reject the project?
11. Payback. A project that costs $2,500 to install will provide annual cash flows of $600 for the next 6 years. The firm accepts projects with payback periods of less than 5 years. Will the project be accepted? Should this project be pursued if the discount rate is 2 percent? What if the discount rate is 12 percent? Will the firm's decision change as the discount rate changes?
12. Profitability Index. What is the profitability index of a project that costs $10,000 and provides cash flows of $3,000 in Years 1 and 2 and $5,000 in Years 3 and 4? The discount rate is 10 percent.
13. NPV. A proposed nuclear power plant will cost $2.2 billion to build and then will produce cash flows of $300 million a year for 15 years. After that period (in Year 15), it must be decommissioned at a cost of $900 million. What is project NPV if the discount rate is 6 percent? What if it is 16 percent?
Practice Problems
14. NPV/IRR. Consider projects A and B: Cash Flows, Dollars Project Cq C
NPV at 10%
21,000 33,000
21,000 33,000
Calculate IRRs for A and B. Which project does the IRR rule suggest is best? Which project is really best?
15. IRR. You have the chance to participate in a project that produces the following cash flows:
M61J
192 part two Value
C0_C_C2
The internal rate of return is 13.6 percent. If the opportunity cost of capital is 12 percent, would you accept the offer? 16. NPV/IRR.
a. Calculate the net present value of the following project for discount rates of 0, 50, and 100 percent:
Co Ci C2
-$6,750 +$4,500 +$18,000 b. What is the IRR of the project?
17. IRR. Marielle Machinery Works forecasts the following cash flows on a project under consideration. It uses the internal rate of return rule to accept or reject projects. Should this project be accepted if the required return is 12 percent?
C0_C_C2_C3
18. NPV/IRR. A new computer system will require an initial outlay of $20,000 but it will increase the firm's cash flows by $4,000 a year for each of the next 8 years. Is the system worth installing if the required rate of return is 9 percent? What if it is 14 percent? How high can the discount rate be before you would reject the project?
19. Investment Criteria. If you insulate your office for $1,000, you will save $100 a year in heating expenses. These savings will last forever.
a. What is the NPV of the investment when the cost of capital is 8 percent? 10 percent?
b. What is the IRR of the investment?
c. What is the payback period on this investment?
20. NPV versus IRR. Here are the cash flows for two mutually exclusive projects: Project_Co_C_C2_C3
A -$20,000 +$8,000 +$8,000 +$8,000 B -$20,000_0_0 +$25,000
a. At what interest rates would you prefer project A to B? Hint: Try drawing the NPV profile of each project.
b. What is the IRR of each project?
21. Payback and NPV. A project has a life of 10 years and a payback period of 10 years. What must be true of project NPV?
22. IRR/NPV. Consider this project with an internal rate of return of 13.1 percent. Should you accept or reject the project if the discount rate is 12 percent?
Year Cash Flow
Payback and NPV.
a. What is the payback period on each of the following projects?
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Fundamentals of Corporate Other Investment Criteria Companies, 2001
Finance, Third Edition chapter 6 Net Present Value and Other Investment Criteria 193
Cash Flows, Dollars
b. Given that you wish to use the payback rule with a cutoff period of 2 years, which projects would you accept?
c. If you use a cutoff period of 3 years, which projects would you accept?
d. If the opportunity cost of capital is 10 percent, which projects have positive NPVs?
e. "Payback gives too much weight to cash flows that occur after the cutoff date." True or false?
24. Book Rate of Return. Consider these data on a proposed project: Original investment = $200 Straight-line depreciation of $50 a year for 4 years Project life = 4 years
Book value |
$200 |
_ |
— |
— |
Sales |
100 |
110 |
120 |
130 |
Costs |
30 |
35 |
40 |
45 |
Depreciation |
— |
— |
— |
_ |
Net income |
— |
— |
_ |
_ |
a. Fill in the blanks in the table.
b. Find the book rate of return of this project in each year.
c. Find project NPV if the discount rate is 20 percent.
25. Book Rate of Return. A machine costs $8,000 and is expected to produce profit before depreciation of $2,500 in each of Years 1 and 2 and $3,500 in each of Years 3 and 4. Assuming that the machine is depreciated at a constant rate of $2,000 a year and that there are no taxes, what is the average return on book?
0 26. Book Rate of Return. A project requires an initial investment of $10,000, and over its 5-year life it will generate annual cash revenues of $5,000 and cash expenses of $2,000. The firm will use straight-line depreciation, but it does not pay taxes.
a. Find the book rates of return on the project for each year.
b. Is the project worth pursuing if the opportunity cost of capital is 8 percent?
c. What would happen to the book rates of return if half the initial $10,000 outlay were treated as an expense instead of a capital investment? Hint: Instead of depreciating all of the $10,000, treat $5,000 as an expense in the first year.
d. Does NPV change as a result of the different accounting treatment proposed in (c)?
27. Profitability Index. Consider the following projects:
Project_Co_C_C2
A -$2,100 +$2,000 +$1,200 B - 2,100 + 1,440 + 1,728
a. Calculate the profitability index for A and B assuming a 20 percent opportunity cost of capital.
b. Use the profitability index rule to determine which project(s) you should accept (i) if you could undertake both and (ii) if you could undertake only one.
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Fundamentals of Corporate Other Investment Criteria Companies, 2001
Finance, Third Edition
28. Capital Rationing. You are a manager with an investment budget of $8 million. You may invest in the following projects. Investment and cash-flow figures are in millions of dollars.
Discount Annual Project Life,
Project Rate, % Investment Cash Flow Years
Discount Annual Project Life,
Project Rate, % Investment Cash Flow Years
A |
10 |
3 |
1 |
5 |
B |
12 |
4 |
1 |
8 |
C |
8 |
5 |
2 |
4 |
D |
8 |
3 |
1.5 |
3 |
E |
12 |
3 |
1 |
6 |
a. Why might these projects have different discount rates?
b. Which projects should the manager choose?
c. Which projects will be chosen if there is no capital rationing?
29. Profitability Index versus NPV. Consider these two projects:
Project |
Co |
Ci |
C2 |
C3 |
A |
-$18 |
+$10 |
+$10 |
+$10 |
B |
-$50 |
+$25 |
+$25 |
+$25 |
a. Which project has the higher NPV if the discount rate is 10 percent?
b. Which has the higher profitability index?
c. Which project is most attractive to a firm that can raise an unlimited amount of funds to pay for its investment projects? Which project is most attractive to a firm that is limited in the funds it can raise?
30. Mutually Exclusive Investments. Here are the cash flow forecasts for two mutually exclusive projects:
Cash Flows, Dollars
Year Project A Project B
1 30 49
2 50 49
3 70 49
a. Which project would you choose if the opportunity cost of capital is 2 percent?
b. Which would you choose if the opportunity cost of capital is 12 percent?
c. Why does your answer change?
31. Equivalent Annual Cost. A precision lathe costs $10,000 and will cost $20,000 a year to operate and maintain. If the discount rate is 12 percent and the lathe will last for five years, what is the equivalent annual cost of the tool?
32. Equivalent Annual Cost. A firm can lease a truck for 4 years at a cost of $30,000 annually. It can instead buy a truck at a cost of $80,000, with annual maintenance expenses of $10,000. The truck will be sold at the end of 4 years for $20,000. Which is the better option if the discount rate is 12 percent?
33. Multiple IRR. Consider the following cash flows:
a. Confirm that one internal rate of return on this project is (a shade above) 7 percent, and that the other is (a shade below) 34 percent.
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II. Value
6. Net Present Value and Other Investment Criteria
© The McGraw-Hill Companies, 2001
chapter 6 Net Present Value and Other Investment Criteria 195
b. Is the project attractive if the discount rate is 5 percent?
c. What if it is 20 percent? 40 percent?
d. Why is the project attractive at midrange discount rates but not at very high or very low rates?
34. Equivalent Annual Cost. Econo-cool air conditioners cost $300 to purchase, result in electricity bills of $150 per year, and last for 5 years. Luxury Air models cost $500, result in electricity bills of $100 per year, and last for 8 years. The discount rate is 21 percent.
a. What are the equivalent annual costs of the Econo-cool and Luxury Air models?
b. Which model is more cost effective?
c. Now you remember that the inflation rate is expected to be 10 percent per year for the foreseeable future. Redo parts (a) and (b).
35. Investment Timing. You can purchase an optical scanner today for $400. The scanner provides benefits worth $60 a year. The expected life of the scanner is 10 years. Scanners are expected to decrease in price by 20 percent per year. Suppose the discount rate is 10 percent. Should you purchase the scanner today or wait to purchase? When is the best purchase time?
36. Replacement Decision. You are operating an old machine that is expected to produce a cash inflow of $5,000 in each of the next 3 years before it fails. You can replace it now with a new machine that costs $20,000 but is much more efficient and will provide a cash flow of $10,000 a year for 4 years. Should you replace your equipment now? The discount rate is 15 percent.
37. Replacement Decision. A forklift will last for only 2 more years. It costs $5,000 a year to maintain. For $20,000 you can buy a new lift which can last for 10 years and should require maintenance costs of only $2,000 a year.
a. If the discount rate is 5 percent per year, should you replace the forklift?
b. What if the discount rate is 10 percent per year? Why does your answer change?
38. NPV/IRR. Growth Enterprises believes its latest project, which will cost $80,000 to install, will generate a perpetual growing stream of cash flows. Cash flow at the end of this year will be $5,000, and cash flows in future years are expected to grow indefinitely at an annual rate of 5 percent.
a. If the discount rate for this project is 10 percent, what is the project NPV?
b. What is the project IRR?
39. Investment Timing. A classic problem in management of forests is determining when it is most economically advantageous to cut a tree for lumber. When the tree is young, it grows very rapidly. As it ages, its growth slows down. Why is the NPV-maximizing rule to cut the tree when its growth rate equals the discount rate?
40. Multiple IRRs. Strip Mining Inc. can develop a new mine at an initial cost of $5 million. The mine will provide a cash flow of $30 million in 1 year. The land then must be reclaimed at a cost of $28 million in the second year.
a. What are the IRRs of this project?
b. Should the firm develop the mine if the discount rate is 10 percent? 20 percent? 350 percent? 400 percent?
Solutions to
Self-test
Questions
6.1 Even if construction costs are $355,000, NPV is still positive:
Therefore, the project is still worth pursuing. The project is viable as long as construction costs are less than the PV of the future cash flow, that is, as long as construction costs are
Brealey-Myers: Fundamentals of Corporate Finance, Third Edition
II. Value
6. Net Present Value and Other Investment Criteria
© The McGraw-Hill Companies, 2001
196 part two Value less than $357,143. However, if the opportunity cost of capital is 20 percent, the PV of the $400,000 sales price is lower and NPV is negative:
The present value of the future cash flow is not as high when the opportunity cost of capital is higher. The project would need to provide a higher payoff in order to be viable in the face of the higher opportunity cost of capital. 6.2 The IRR is now about 8.9 percent because
$16,000
Note in Figure 6.6 that NPV falls to zero as the discount rate reaches 8.9 percent.
6.3 The payback period is $5,000/$660 = 7.6 years. Calculate NPV as follows. The present value of a $660 annuity for 20 years at 6 percent is
PV annuity = $7,570 NPV = -$5,000 + $7,570 = +$2,570
The project should be accepted.
6.4 a. IRR = 23% (i.e., -60 + 30/1.23 + 30/1.232 + 30/1.233 = 0).
b. Year 1: Income/book value at start of Year 1 = (30 - 20)/60 = .17, or 17%. Year 2: Income/book value at start of Year 2 = (30 - 20)/40 = .25, or 25%. Year 3: Income/book value at start of Year 3 = (30 - 20)/20 = .50, or 50%.
6.5 Year of Purchase
Cost of Computer
PV Savings
NPV at Year of Purchase
NPV Today
6.5 Year of Purchase
Cost of Computer
PV Savings
NPV at Year of Purchase
NPV Today
0 |
50 |
70 |
20 |
20 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1 |
45 |
66 |
21 |
19.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
2 |
40 |
62 |
22 |
18.2 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
3 |
36 |
58 |
22 |
16.5 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
4 |
33 |
54 |
21 |
14.3 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
5 |
31 |
50 |
19 |
NPV falls to zero at an interest rate of 8.9 percent. -50 100 150 200 250 12 16 20 24 28 32 Discount rate, percent 36 40 44 48 chapter 6 Net Present Value and Other Investment Criteria 197
Machine G is the better buy. However, it's still better to keep the old machine going one more year. That costs $4,300, which is less than G's equivalent annual cost, $6,019. 6.7 You want to be rich. The NPV of the long-lived investment is much larger. 6.8 You want to be richer. The second alternative generates greater value at any reasonable discount rate. For example, suppose other risk-free investments offer 8 percent. Then $1,500,000 1.08 6.9 Rank each project in order of profitability index as in the following table:
Starting from the top, we run out of funds after accepting projects N and L. While L and O have equal profitability indexes, project O could not be chosen because it would force total investment above the limit of $10 million. 6.10 The profitability index gives the wrong ranking for the first pair, correct ranking for the second: Profitability Index Project_pV_Investment_NPV_(NPV/Investment) Profitability Index Project_pV_Investment_NPV_(NPV/Investment)
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