罗斯公司理财第六版习题答案第6章

Concept Questions - Chapter 6

6.2  List the problems of the payback period rule.

1. It does not take into account the time value of money. 2. It ignores payments after the payback period. 3. The cutoff period is arbitrary.  What are some advantages?

1. It is simple to implement.

2. It may help in controlling and evaluating managers. 6.4  What are the three steps in calculating AAR?

1. Determine average net income. 2. Determine average investment

3. Divide average net income by average investment.  What are some flaws with the AAR approach?

1. It uses accounting figures. 2. It takes no account of timing. 3. The cutoff period is arbitrary.

6.5  How does one calculate the IRR of a project?

Using either trial-and-error or a financial calculator, one finds the discount rate that produces an NPV of zero.

6.6  What is the difference between independent projects and mutually exclusive projects?

An independent project is one whose acceptance does not affect the acceptance of another. A mutually exclusive project, on the other hand is one whose acceptance precludes the acceptance of another.

 What are two problems with the IRR approach that apply to both

independent and mutually exclusive projects?

1. The decision rule depends on whether one is investing of financing. 2. Multiple rates of return are possible.

 What are two additional problems applying only to mutually exclusive

projects?

1. The IRR approach ignores issues of scale.

2. The IRR approach does not accommodate the timing of the cash flows

properly.

6.7  How does one calculate a project's profitability index?

Divide the present value of the cash flows subsequent to the initial investment by the initial investment.

 How is the profitability index applied to independent projects, mutually

exclusive projects, and situations of capital rationing?

1. With independent projects, accept the project if the PI is greater than 1.0 and

reject if less than 1.0.

2. With mutually exclusive projects, use incremental analysis, subtracting the

cash flows of project 2 from project 1. Find the PI. If the PI is greater than 1.0, accept project 1. If less than 1.0, accept project 2.

3. In capital rationing, the firm should simply rank the projects according to their

respective PIs and accept the projects with the highest PIs, subject to the budget constrain.

Answers to End-of-Chapter Problems

QUESTIONS AND PROBLEMS The Payback Period Rule

6.1 Fuji Software, Inc., has the following projects. Year Project A Project B 0 _$7,500 _$5,000 1 4,000 2,500 2 3,500 1,200 3 1,500 3,000

a. Suppose Fuji’s cutoff payback period is two years. Which of these two projects should be chosen? b. Suppose Fuji uses the NPV rule to rank these two projects. If the appropriate discount rate is 15 percent, which project should be chosen?

6.1

a. b. Payback period of Project A = 1 + ($7,500 - $4,000) / $3,500 = 2 years

Payback period of Project B = 2 + ($5,000 - $2,500 -$1,200) / $3,000 = 2.43 years Project A should be chosen.

NPVA = -$7,500 + $4,000 / 1.15 + $3,500 / 1.152 + $1,500 / 1.153 = -$388.96 NPVB = -$5,000 + $2,500 / 1.15 + $1,200 / 1.152 + $3,000 / 1.153 = $53.83 Project B should be chosen.

6.2 Suppose Peach Paving Company invests $1 million today on a new construction project. The project will generate annual cash flows of $150,000 in perpetuity. The appropriate annual discount rate for the project is 10 percent.

a. What is the payback period for the project? If the Peach Paving Company desires to have a 10-year payback period, should the project be adopted?

b. What is the discounted payback period for the project? c. What is the NPV of the project?

6.2 a. b. c. Payback period = 6 + {$1,000,000 - ($150,000  6)} / $150,000 = 6.67 years Yes, the project should be adopted. $150,000 110.10 = $974,259

The discounted payback period = 11 + ($1,000,000 - $974,259) / ($150,000 / 1.112)

= 11.54 years

NPV = -$1,000,000 + $150,000 / 0.10 = $500,000

The Average Accounting Return

6.3 The annual, end-of-year, book-investment accounts for the machine whose purchase your firm is considering are shown below. Purchase Year Year Year Year Date 1 2 3 4

Gross investment $16,000 $16,000 $16,000 $16,000 $16,000 Less: accumulated

depreciation ______0_ ___4_,0_0_0_ ___8_,0_0_0_ __1_2_,0_0_0_ _1_6_,_0_0_0 Net investment $16,000 $12,000 $ 8,000 $ 4,000 $ 0

If your firm purchases this machine, you can expect it to generate, on average, $4,500 per year in additional net income.

a. What is the average accounting return for this machine? b. What three flaws are inherent in this decision rule?

6.3 a. Average Investment:

($16,000 + $12,000 + $8,000 + $4,000 + 0) / 5 = $8,000 Average accounting return:

$4,500 / $8,000 = 0.5625 = 56.25%

b. 1. 2. 3. AAR does not consider the timing of the cash flows, hence it does not consider the time value of money.

AAR uses an arbitrary firm standard as the decision rule. AAR uses accounting data rather than net cash flows.

6.4 Western Printing Co. has an opportunity to purchase a $2 million new printing machine. It has an economic life of five years and will be worthless after that time. This new investment is expected to generate an annual net income of $100,000 one year from today and the income stream will grow at 7

percent per year subsequently. The company adopts a straight-line depreciation method (i.e., equal amounts of depreciation in each year). What is the average accounting return of the investment? Supposing Western Printing’s AAR cutoff is 20 percent, should the machine be purchased?

6.4 Average Investment = ($2,000,000 + 0) / 2 = $1,000,000 Average net income = [$100,000 {(1 + g)5 - 1} / g] / 5 = {$100,000A (1.075 - 1} / 0.07} / 5 = $115,014.78

AAR = $115,014.78 / $1,000,000 = 11.50%

No, since the machine’s AAR is less than the firm’s cutoff AAR.

6.5 Nokia Group has invested $8,000 in a high-tech project. This cost is depreciated on an accelerated basis that yields $4,000, $2,500, $1,500 of depreciation, respectively, during its three-year economic life. The project is expected to produce income before tax of $2,000 each year during its economic life. If the tax rate is 25%, what is the project’s average accounting return (AAR)? a. 44.44% b. 50.23% c. 66.67% d. 70.00% e. 82.21%

The Internal Rate of Return

6.5 a

6.6 Compute the internal rate of return on projects with the following cash flows. Cash Flows ($)

Year Project A Project B 0 _3,000 _6,000 1 2,500 5,000 2 1,000 2,000

6.6

PI = $40,000 70.15 / $160,000 = 1.04

Since the PI exceeds one accept the project.

6.7 CPC, Inc., has a project with the following cash flows. Year Cash Flows ($) 0 _8,000 1 4,000 2 3,000 3 2,000

a. Compute the internal rate of return on the project.

b. Suppose the appropriate discount rate is 8 percent. Should the project be adopted by CPC?

6.7 The IRR is the discount rate at which the NPV = 0. -$3,000 + $2,500 / (1 + IRRA) + $1,000 / (1 + IRRA)2 = 0 By trial and error, IRRA = 12.87% Since project B’s cash flows are two times of those of project A, the IRRB = IRRA =

12.87%

6.8 Compute the internal rate of return for the cash flows of the following two projects. Cash Flows ($) Time A B

0 _2,000 _1,500 1 2,000 500

2 8,000 1,000 3 _8,000 1,500

6.8 a. b. Solve x by trial and error:

-$4,000 + $2,000 / (1 + x) + $1,500 / (1 + x)2 + $1,000 / (1 + x)3 = 0 x = 6.93%

No, since the IRR (6.93%) is less than the discount rate of 8%.

6.9 Suppose you are offered $5,000 today and obligated to make scheduled payments as follows: Year Cash Flows ($) 0 5,000 1 _2,500 2 _2,000 3 _1,000 4 _1,000

a. What is the IRRs of this offer?

b. If the appropriate discount rate is 10 percent, should you accept this offer? c. If the appropriate discount rate is 20 percent, should you accept this offer? Chapter 6 Some Alternative Investment Rules 165

d. What is the corresponding NPV of the project if the appropriate discount rates are 10 percent and 20 percent, respectively? Are the choices under the NPV rule consistent with those of the IRR rule?

6.9 Find the IRRs of project A analytically. Since the IRR is the discount rate that makes the NPV

equal to zero, the following equation must hold.

-$200 + $200 / (1 + r) + $800 / (1 + r)2 - $800 / (1 + r)3 = 0 $200 [-1 + 1 / (1 + r)] - {$800 / (1 + r)2}[-1 + 1 / (1 + r)] = 0 [-1 + 1 / (1 + r)] [$200 - $800 / (1 + r)2] = 0

For this equation to hold, either [-1 + 1 / (1 + r)] = 0 or [$200 - $800 / (1 + r)2] = 0. Solve each of these factors for the r that would cause the factor to equal zero. The resulting rates are the two IRRs for project A. They are either r = 0% or r = 100%. Note: By inspection you should have known that one of the IRRs of project A is zero. Notice that the sum of the un-discounted cash flows for project A is zero. Thus, not discounting the cash flows would yield a zero NPV. The discount rate which is tantamount to not discounting is zero. Here are some of the interactions used to find the IRR by trial and error. Sophisticated calculators can compute this rate without all of the tedium involved in the trial-and-error method. NPV = -$150 + $50 / 1.3 + $100 / 1.32 + $150 / 1.33 = $15.91 NPV = -$150 + $50 / 1.4 + $100 / 1.42 + $150 / 1.43 = -$8.60 NPV = -$150 + $50 / 1.37 + $100 / 1.372 + $150 / 1.373 = -$1. NPV = -$150 + $50 / 1.36 + $100 / 1.36 2 + $150 / 1.363 = $0.46 NPV = -$150 + $50 / 1.36194 + $100 / 1.361942 + $150 / 1.361943 = $0.0010 NPV = -$150 + $50 / 1.36195 + $100 / 1.361952 + $150 / 1.361953 = -$0.0013 NPV = -$150 + $50 / 1.361944 + $100 / 1.3619442 + $150 / 1.3619443 = $0.0000906 Thus, the IRR is approximately 36.1944%.

6.10 As the Chief Financial Officer of the Orient Express, you are offered the following two mutually exclusive projects. Year Project A Project B 0 _$5,000 _$100,000 1 3,500 65,000 2 3,500 65,000

a. What are the IRRs of these two projects?

b. If you are told only the IRRs of the projects, which would you choose?

c. What did you ignore when you made your choice in part (b)? d. How can the problem be remedied?

e. Compute the incremental IRR for the projects.

f. Based on your answer to part (e), which project should you choose?

g. Suppose you have determined that the appropriate discount rate for these projects is 15 percent. According to the NPV rule, which of these two projects should be adopted?

6.10

a. b. c. d. Solve r in the equation:

$5,000 - $2,500 / (1 + r) - $2,000 / (1 + r)2 - $1,000 / (1 + r)3 - $1,000 / (1 + r)4 = 0 By trial and error, IRR = r = 13.99%

Since this problem is the case of financing, accept the project if the IRR is less than the required rate of return. IRR = 13.99% > 10% Reject the offer.

IRR = 13.99% < 20% Accept the offer. When r = 10%:

NPV = $5,000 - $2,500 / 1.1 - $2,000 / 1.12 - $1,000 / 1.13 - $1,000 / 1.14

= -$359.95 When r = 20%:

NPV = $5,000 - $2,500 / 1.2 - $2,000 / 1.22 - $1,000 / 1.23 - $1,000 / 1.24

= $466.82

Yes, they are consistent with the choices of the IRR rule since the signs of the cash flows change only once.

6.11 Consider two streams of cash flows, A and B. Cash flow A consists of $5,000 starting three years from today and growing at 4 percent in perpetuity. Cash flow B consists of _$6,000 starting two years from today and continuing in perpetuity. Assume the appropriate discount rate is 12 percent. a. What is the present value of each stream?

b. What is the IRR of a project C, which is a combination of projects A and B; that is, C _ A _ B? c. If it is assumed that the discount rate is always positive, what is the rule related to IRR for assessing project C that would correspond to the NPV rule?

6.11

a. b. c. d. e.

Project A:

NPV = -$5,000 + $3,500 / (1 + r) + $3,500 / (1 + r)2 = 0 IRR = r = 25.69% Project B:

NPV = -$100,000 + $65,000 / (1 + r) + $65,000 / (1 + r)2 = 0 IRR = r = 19.43%

Choose project A because it has a higher IRR. The difference in scale is ignored. Apply the incremental IRR method. C0 C1 C2 B - A -$95,000 $61,500 $61,500 NPV = -$95,000 + $61,500 / (1 + r) + $61,500 / (1 + r)2 = 0 Incremental IRR = r = 19.09%

If the discount rate is less than 19.09%, choose project B. Otherwise, choose project A.

NPVA = -$5,000 + $3,500 / 1.15 + $3,500 / 1.152 = $6.98

NPVB = -$100,000 + $65,000 / 1.15 + $65,000 / 1.152 = $5,671.08 Choose project B.

f. g.

6.12 Project A involves an investment of $1 million, and project B involves an investment of $2 million. Both projects have a unique internal rate of return of 20 percent. Is the following statement true or false? Explain your answer.

For any discount rate between 0 percent and 20 percent, inclusive, project B has an NPV twice as great as that of project A.

6.12 a. b. c.

PVA = {$5,000 / (0.12 - 0.04)} / 1.122 = $49,824.61 PVB = (-$6,000 / 0.12) / 1.12 = -$44,2.86 The IRR for project C must solve

{$5,000 / (x - 0.04)} / (1 + x)2 + (-$6,000 / x) / (1 + x) = 0 $5,000 / (x - 0.04) - $6,000 (1 + x) / x = 0 25 x2 + 3.17 x - 1 =0

x = {-3.17 - (110.04)0.5} / 50 or {-3.17 + (110.04)0.5} / 50 The relevant positive root is IRR = x = 0.14 = 14.%

To arrive at the appropriate decision rule, we must graph the NPV as a function of the discount rate. At a discount rate of 14.% the NPV is zero. To determine if the graph is upward or downward sloping, check the NPV at another discount rate. At a discount rate of 10% the NPV is $14,325.07 [= $68,870.52 - $54,545.54]. Thus, the graph of the NPV is downward sloping. From the discussion in the text, if an NPV graph is downward sloping, the project is an investing project. The correct decision rule for an investing project is to accept the project if the discount rate is below 14.%.

NPV

10%

$14,325.07

0

r

14.%

The Profitability Index

6.13 Suppose the following two mutually exclusive investment opportunities are available to the DeAngelo Firm. The appropriate discount rate is 10 percent. Year Project Alpha Project Beta 0 _$500 _$2,000 1 _300 _300 2 700 1,800 3 600 1,700

a. What is the NPV of project alpha and project beta?

b. Which project would you recommend for the DeAngelo Firm?

6.13 Generally, 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. Examples are easy to construct.

C0 C1 C2 IRR NPV @ 0%

A: -$1,000,000 $0 $1,440,000 0.20 $440,000 B: -2,000,000 2,400,000 0 0.20 400,000

In one particular case, the statement is true for equally risky projects. If the lives of the two projects are equal and in every time period the cash flows of the project B are twice the cash flows of project A, then the NPV of project B will be twice as great as the NPV of project A for any discount rate between 0% and 20%.

6.14 The firm for which you work must choose between the following two mutually exclusive projects. The appropriate discount rate for the projects is 10 percent. Profitability

C0 C1 C2 Index NPV

A _$1,000 $1,000 $500 1.32 $322 B _500 500 400 1.57 285

The firm chose to undertake A. At a luncheon for shareholders, the manager of a pension fund that owns a substantial amount of the firm’s stock asks you why the firm chose project A instead of project B when B is more profitable.

How would you justify your firm’s action? Are there any circumstances under which the pension fund manager’s argument could be correct?

6.14 a. b. NPV = $756.57 - $500 = $256.57

NPV = $2,492.11 - $2,000 = $492.11 Choose project beta.

6.15 The treasurer of Davids, Inc., has projected the cash flows of projects A, B, and C as follows. Suppose the relevant discount rate is 12 percent a year. Year Project A Project B Project C 0 _$100,000 _$200,000 _$100,000 1 70,000 130,000 75,000 2 70,000 130,000 60,000

a. Compute the profitability indices for each of the three projects. b. Compute the NPVs for each of the three projects.

c. Suppose these three projects are independent. Which projects should Davids accept based on the profitability index rule?

d. Suppose these three projects are mutually exclusive. Which project should Davids accept based on the profitability index rule?

e. Suppose Davids’ budget for these projects is $300,000. The projects are not divisible. Which projects should Davids accept?

6.15 Although the profitability index is higher for project B than for project A, the NPV is the

increase in the value of the company that will occur if a particular project is undertaken. Thus, the project with the higher NPV should be chosen because it increases the value of the firm the most. Only in the case of capital rationing could the pension fund manager be correct.

6.16 Bill plans to open a self-serve grooming center in a storefront. The grooming equipment will cost $160,000. Bill expects the after-tax cash inflows to be $40,000 annually for seven years, after which he plans to scrap the equipment and retire to the beaches of Jamaica.

Assume the required return is 15%. What is the project’s PI? Should it be accepted? Comparison of Investment Rules

6.16 a. b. c.

PIA = ($70,000 / 1.12 + $70,000 / 1.122) / $100,000 = 1.183 PIB = ($130,000 / 1.12 + $130,000 / 1.122) / $200,000 = 1.099 PIC = ($75,000 / 1.12 + $60,000 / 1.122) / $100,000 = 1.148 NPVA = -$100,000 + $118,303.57 = $18,303.57 NPVB = -$200,000 + $219,706.63 = $19,706.63 NPVC = -$100,000 + $114,795.92 = $14,795.92

Accept all three projects because PIs of all the three projects are greater than one.

d.

Based on the PI rule, project C can be eliminated because its PI is less than the one of project A, while both have the same amount of the investment. We can compute the PI of the incremental cash flows between the two projects, Project C0 C1 C2 PI B - A -$100,000 $60,000 $60,000 1.014

We should take project B since the PI of the incremental cash flows is greater than one.

Project B has the highest NPV, while A has the next highest NPV. Take both projects A and B.

e.

6.17 Define each of the following investment rules. In your definition state the criteria for accepting or rejecting an investment under each rule. a. Payback period

b. Average accounting return c. Internal rate of return d. Profitability index e. Net present value

6.17 a.

b.

c.

d.

e.

The payback period is the time it takes to recoup the initial investment of a project. Accept any project that has a payback period that is equal to or shorter than the company’s standard payback period. Reject all other projects. The average accounting return (AAR) is defined as

Average project earnings  Average book value of the investment.

Accept projects for which the AAR is equal to or greater than the firm’s standard. Reject all other projects.

The internal rate of return (IRR) is the discount rate which makes the net present value (NPV) of the project zero. The accept / reject criteria is:

If C0 < 0 and all future cash flows are positive, accept the project if IRR  discount rate.

If C0 < 0 and all future cash flows are positive, reject the project if IRR < discount rate.

If C0 > 0 and all future cash flows are negative, accept the project if IRR  discount rate.

If C0 > 0 and all future cash flows are negative, reject the project if IRR > discount rate.

If the project has cash flows that alternate in sign, there is likely to be more than one positive IRR. In that situation, there is no valid IRR accept / reject rule. The profitability index (PI) is defined as:

(The present value of the cash flows subsequent to the initial investment  The initial investment)

Accept any project for which the profitability index is equal to or greater than one. Reject project for which that is not true.

The net present value (NPV) is the sum of the present values of all project cash flows. Accept those projects with NPVs which are equal to or greater than zero. Rejects proposals with negative NPVs.

6.18 Consider the following cash flows of two mutually exclusive projects for Chinese Daily News. New Sunday New Saturday Year Early Edition Late Edition 0 _$1,200 _$2,100 1 600 1,000 2 550 900 3 450 800

a. Based on the payback period rule, which project should be chosen?

b. Suppose there is no corporate tax and the cash flows above are income before the depreciation. The firm uses a straight-line depreciation method (i.e., equal amounts of depreciation in each year). What is the average accounting return for each of these two projects? c. Which project has a greater IRR?

d. Based on the incremental IRR rule, which project should be chosen?

6.18 Let project A represent New Sunday Early Edition; and let project B represent New

Saturday Late Edition.

a. Payback period of project A = 2 + ($1,200 - $1,150) / $450 = 2.11 years Payback period of project B = 2 + ($2,100 - $1,900) / $800 = 2.25 years Based on the payback period rule, you should choose project A. b. Project A: Average investment = ($1,200 + $0) / 2 = $600 Depreciation = $400 / year Average income = [($600 - $400) + ($550 - $400) + ($450 - $400)] / 3 = $133.33 AAR = $133.33 / $600 = 22.22% Project B: Average investment = ($2,100 + $0) / 2 = $1,050 Depreciation = $700 / year Average income = [($1,000 - $700) + ($900 - $700) + ($800 - $700)] / 3 = $200 AAR = $200 / $1,050 = 19.05% c. IRR of project A: -$1,200 + $600 / (1 + r) + $550 / (1 + r)2 + $450 / (1 + r)3 = 0 IRR = r = 16.76% IRR of project B: -$2,100 + $1,000 / (1 + r) + $900 / (1 + r)2 + $800 / (1 + r)3 = 0 IRR = r = 14.29% Project A has a greater IRR.

d. IRR of project B-A: Incremental cash flows

Year 0 1 2 3

B - A -$900 $400 $350 $350

23

-$900 + $400 / (1 + r) + $350 / (1 + r) + $350 / (1 + r) = 0 Incremental IRR = r = 11.02%

If the required rate of return is greater than 11.02%, then choose project A. If the required rate of return is less than 11.02%, then choose project B.

6.19 Consider the following cash flows on two mutually exclusive projects that require an annual return of 15 percent. Working in the financial planning department for the Bahamas Recreation Corp., you are trying to compare different investment criteria to arrive at a sensible choice of these two projects. Deepwater New Submarine Year Fishing Ride

0 _$600,000 _$1,800,000 1 270,000 1,000,000 2 350,000 700,000 3 300,000 900,000

a. Based on the discounted payback period rule, which project should be chosen?

b. If your decision rule is to accept the project with a greater IRR, which project should you choose? c. Since you are fully aware of the IRR rule’s scale problem, you calculate the incremental IRR for the cash flows. Based on your computation, which project should you choose?

d. To be prudent, you compute the NPV for both projects. Which project should you choose? Is it consistent with the incremental IRR rule?

6.19 Let project A be Deepwater Fishing; let project B be New Submarine Ride. a. Project A:

Year Discounted CF Cumulative CF 0 -$600,000 -$600,000 1 234,783 -365,217 2 2,650 -100,567 3 197,255

Discounted payback period of project A = 2 + $100,567 / $197,255 = 2.51 years Project B:

Year Discounted CF Cumulative CF 0 -$1,800,000 -$1,800,000 1 869,565 -930,435 2 529,301 -401,134 3 591,765

Discounted payback period of project B = 2 + $401,134 / $591,765 = 2.68 years Project A should be chosen. b. IRR of project A: -$600,000 + $270,000 / (1 + r) + $350,000 / (1 + r)2 + $300,000 / (1 + r)3 = 0 IRR = r = 24.30% IRR of project B: -$1,800,000 + $1,000,000 /(1 + r) + $700,000 / (1 + r)2 + $900,000 / (1 + r)3

= 0 IRR = r = 21.46% Based on the IRR rule, project A should be chosen since it has a greater IRR. c. Incremental IRR:

Year 0 1 2 3

d.

B - A -$1,200,000 $730,000 $350,000 $600,000

2

-$1,200,000 + $730,000 / (1 + r) + $350,000 / (1 + r) + $600,000 / (1 + r)3 = 0 Incremental IRR = r = 19.92%

Since the incremental IRR is greater than the required rate of return, 15%, choose project B.

NPVA = -$600,000 + $270,000 / 1.15 + $350,000 / 1.152 + $300,000 / 1.153

= $96,687.76

NPVB = -$1,800,000 + $1,000,000 / 1.15 + $700,000 / 1.152 + $900,000 / 1.153

= $190,630.39

Since NPVB > NPVA, choose project B.

Yes, the NPV rule is consistent with the incremental IRR rule.

6.20 The Utah Mining Corporation is set to open a gold mine near Provo, Utah. According to the treasurer, Steven Sample, “This is a golden opportunity.” The mine will cost $600,000 to open. It will generate a cash inflow of $100,000 during the first year and the cash flows are projected to grow at 8 percent per year for 10 years. After 10 years the mine will be abandoned. Abandonment costs will be $50,000. a. What is the IRR for the gold mine?

b. The Utah Mining Corporation requires a 10 percent return on such undertakings. Should the mine be opened?

6.20 a. The IRR is the discount rate at which the NPV = 0

b.

11$100,00018%$50,0000 -$600,000 + 111(r8%)1r1rIRR 18.56%

Yes, the mine should be opened since its IRR exceeds its required return of 10%.