Fundamentals of corporate finance brealey chapter 07 NPV and other investment criteria

Since the discounted cash flows are negative until year 3 and become positive by Year 4, the project pays back sometime in the fourth year.. Despite its higher payback, Project A still m

Solutions to Chapter 7 NPV and Other Investment Criteria

1 NPVA = –100 + 40  annuity factor(11%, 4 periods) = $24.10

NPVB = –100 + 50  annuity factor(11%, 3 periods) = $22.19

Both projects are worth pursuing

2 Choose the project with the higher NPV, project A

3 If r = 16%, then NPVA = $11.93 and NPVB = $12.29 Therefore, you should now choose project B

4 IRRA = Discount rate at which 40  annuity factor(r, 4 periods) = 100

IRRA = 21.86%

IRRB = 23.38%

5 No Even though project B has the higher IRR, its NPV is lower than that of project

A when the discount rate is lower (as in Problem 1) and higher when the discount rate is higher (as in Problem 3) This example shows that the project with the higher IRR is not necessarily better The IRR of each project is fixed, but as the

discount rate increases, project B becomes relatively more attractive compared to

project A This is because B’s cash flows come earlier, so their present values fall less rapidly when the discount rate increases

6 The profitability indexes are as follows:

Project A 24.10/100 = 2410

Project B 22.19/100 = 2219

In this case, with equal initial investments, both the profitability index and NPV

will give projects the same ranking This is an unusual case, however, since it is rare for initial investments to be equal

7 Project A has a payback period of 100/40 = 2.5 years Project B has a payback period of 2 years

8 Project A

Year Cash Flow ($) Discounted Cash Flow($) @ 11 percent Cumulative DiscountedCash Flow ($)

Assuming uniform cash flows across time, the fractional year can now be

determined Since the discounted cash flows are negative until year 3 and become positive by Year 4, the project pays back sometime in the fourth year Note that out

of the total discounted cash flow of $26.36 in Year 4, the first $2.24 comes in by 2.24/26.36 = 0.084 year Therefore, the discounted payback period for Project A is 3.084 years

Project B

Year Cash Flow ($) Discounted Cash Flow

($) @ 11 percent

Cumulative DiscountedCash Flow ($)

The discounted payback for Project B is 2 years + 14.35/36.55 = 2.39 years

9 No Despite its higher payback, Project A still may be the preferred project, for example, when the discount rate is 11% (as in Problems 1 and 2) Just as in

problem 5, you should note that the payback period for each project is fixed, but that NPV changes as the discount rate changes The project with the shorter

payback period need not have the higher NPV

10 NPV = 3,000 + 800  annuity factor(10%, 6 years) = $484.21

At this discount rate, you should accept the project

You can solve for IRR by setting the PV of cash flows equal to 3,000 on your calculator and solving for the interest rate: PV = 3000; n = 6; FV = 0; PMT = 800; compute i The IRR is 15.34%, which is the highest discount rate before project NPV turns negative

11 Payback = 2500/600 = 4.167 years, which is less than the cutoff So the firm wouldaccept the project

12 NPV = 10,000 + + + + = $2,378.25

Profitability index = NPV/Investment = 2378

13 Project at 2 percent discount rate

Year Cash Flow ($) Discounted Cash Flow($) @ 2 percent Cumulative DiscountedCash Flow ($)

$739.20 in Year 4, the first $693.60 comes in by 693.60/739.20 = 0.94 year

Therefore, the discounted payback for the project is 3.94 years, and thus the projectshould be pursued

Project at 12 percent discount rate

Year Cash Flow ($) Discounted Cash Flow($) @ 12 percent Cumulative DiscountedCash Flow ($)

Since the discounted cash flows become positive by Year 6, the project pays back

in 5 years + 116/405.6 = 5.28 years Therefore, given the firm’s decision criteria of

a discounted payback of 5 years or less, the project should not be pursued

As illustrated by the two scenarios above, the firm’s decision will change as the discount rate changes As the discount rate increases, the discounted payback periodgets extended

14 NPV = 2.2 + 3  annuity factor(r, 15 years)  9/(1 + r)15

When r = 6%, NPV = 2.2 + 2.538 = $0.338 billion

When r = 16%, NPV = 2.2 + 1.576 = $0.624 billion

15 The IRR of project A is 25.69%, and that of B is 20.69% However, project B has the higher NPV and therefore is preferred The incremental cash flows of B over A are –20,000 at time 0 and +12,000 at times 1 and 2 The NPV of the incremental

cash flows is $827, which is positive and equal to the difference in project NPVs

16 NPV = 5000 + – = –$197.70

Because NPV is negative, you should reject the offer You should reject the offer despite the fact that IRR exceeds the discount rate This is a “borrowing type”

project with positive cash flows followed by negative cash flows A high IRR in

these cases is not attractive: You don’t want to borrow at a high interest rate

17 a r = 0 implies NPV = 6,750 + 4,500 + (-18,000) = $-6,750

r = 50% implies NPV = 6750 + + 2

5.1

000,18

500,7



12.1

500,8



= $-2,029.08which is negative So the project is not attractive

However, you can note that the IRR of the project is 37.03 % Since the IRR of the project is greater than the required rate of return of 12%, the project should be accepted using this rule On balance, we would use the NPV rule and reject the project

19 NPV9% = –20,000 + 4,000  annuity factor(9%, 8 periods)

= $2139.28NPV14% = –20,000 + 4,000  annuity factor(14%, 8 periods)

= –$1,444.54

The IRR is 11.81% To confirm this on your calculator, set PV = ()20,000; PMT = 4000;

FV = 0; n = 8, and compute i The project will be rejected for any discount rate above thisrate

20 a The present value of the savings is 100/r

The NPV=0 when the cost of capital =10% The savings are supposed to last

forever Therefore, there is no finite discounted payback period when cost of

capital is 10%

21 a NPV of the two projects at various discount rates is tabulated below

NPVA = –20,000 + 8,000  annuity factor(r%, 3 years)

From the NPV profile, it can be seen that Project A is preferred over Project

B if the discount rate is above 4% At 4% and below, Project B has the higherNPV

b IRRA = 9.70% [PV = (–)20; PMT = 8; FV = 0; n = 3; compute i]

IRRB = 7.72% [PV = (–)20; PMT = 0; FV = 25; n = 3; compute i]

22 We know that the undiscounted project cash flows must sum to the initial

investment because payback equals project life Therefore, the discounted cash

flows are less than the initial investment, so NPV must be negative

23 NPV = 100 + + = –1.40

Because NPV is negative, you should reject the offer This is so despite the fact thatIRR exceeds the discount rate This is a “borrowing type” project with a positive cash flow followed by negative cash flows A high IRR in these cases is not

attractive: You don’t want to borrow at a high interest rate

The payback period for Project A is 3 years.

Project A does not pay back on a discounted basis since cumulative

discounted cash flows remain negative until the end of Year 4

The payback period for Project B is 2 years

The discounted payback period for Project B is 2 years + 173.55/1502.63 = 2.12 years

The payback period for Project C is 3 years

The discounted payback period for Project C is 3 years + 1010.52/3415.07 = 3.3 years

b Only B satisfies the 2-year payback criterion

c You would accept Project B

26 a Cash flow each year = $5,000 – $2,000 = $3,000

NPV = –10,000 + 3,000  annuity factor(8%, 5 years) = 1,978.13

NPV is positive so you should pursue the project

b The accounting change has no effect on project cash flows, and therefore no effect on NPV

27 a The present values of the project cash flows (net of the initial investments)

are:

NPVA = –2100 + + = $400

NPVB = –2100 + + = $300

The initial investment for each project is 2100

Profitability index (A) = 400/2100 = 0.1905

Profitability index (B) = 300/2100 = 0.1429

b If you can choose only one project, choose A for its higher profitability index.

If you can take both projects, you should: Both have positive profitability

index

28 a The less–risky projects should have lower discount rates

b First, find the profitability index of each project

Then select projects with the highest profitability index until the $8 million budget

is exhausted Choose, therefore, projects E and C

c All the projects have positive NPV All will be chosen if there is no rationing

29 a NPVA = –18 + 10  annuity factor(10%, 3 periods) = $ 6.87

NPVB = –50 + 25  annuity factor(10%, 3 periods) = $12.17

Thus Project B has the higher NPV if the discount rate is 10%

b Project A has the higher profitability index

Invest Profitability IndexProject PV ment NPV (= NPV/Investment)

c A firm with a limited amount of funds available should choose Project A

since it has a higher profitability index of 0.38, i.e., a higher “bang for the

buck.”

For a firm with unlimited funds, the possibilities are:

i If the projects are independent projects, then the firm should choose

both projects

ii However, if the projects are mutually exclusive, then Project B should

be selected It has the higher NPV

Discounted payback period is 2 years + 3729.09/5037.72 = 2.74 years.

Project II has the lower discounted payback period at 2.74 years, and thus it isthe best choice under this decision criterion

(3) NPV at 6% for: Project I = $3031.19

Project II = $6434.82Since Project II offers a higher NPV, it is the best choice under this decision criterion

(4) IRRI = 6.29%

IRRII = 19.04%

Project II has the higher IRR and is, therefore, the better choice

(5) Profitability Index (PI) =

i

Investment Initial

NPV

PI Project I = 0 012

000 , 250

19 031 , 3

PI Project II = 0 26

000 , 25

82 434 , 6

Project II offers the highest ratio of net present value to investment; therefore,

we would choose Project II

b Of the two projects, Project II is the better choice It has lower payback and discounted payback periods In addition, Project II has the higher NPV, IRR and profitability index

32 PV of Costs = 10,000 + 20,000  annuity factor(12%, 5 years)

= 10,000 + 72,096 = $82,096The equivalent annual cost is the payment with the same present value, $22,774.[n = 5, i = 12, FV = 0; PV = ()82,096; compute PMT]

33 Buy: PV of Costs = 80,000 + 10,000  annuity factor(12%, 4 years)  20,000/(1.12)4

= 80,000 + 30,373  12,710 = $97,663The equivalent annual cost is the payment with the same present value, $32,154.[n = 4, i = 12, FV = 0; PV = ()97,663; compute PMT]

If you can lease instead for $30,000, then this is the less costly option

You also can compare the PV of the lease costs to the total PV of buying:

30,000 annuity factor(12%, 4 years) = $91,120

which is less than the PV of costs when buying the truck

34 a The following table shows the NPV profile of the project NPV is zero at an

interest rate between 7% and 8% and at an interest rate between 33% and

34% These are the two IRRs of the project You can use your calculator to confirm that the two IRR’s are, more precisely, 7.16% and 33.67%

Discount rate NPV Discount rate NPV

At very high rates, the positive cash flows are discounted very heavily, resulting in a negative NPV For mid-range discount rates, the positive cash flows that occur in the middle of the project dominate and project NPV is positive

35 a Econo-cool costs $300 and lasts for 5 years The annual rental fee with the

b Luxury Air is more cost effective It has the lower equivalent annual cost

c The real interest rate is now 1.21/1.10 – 1 = 10 = 10%

Redo (a) and (b) using a 10% discount rate Because energy costs would normally be expected to inflate along with all other costs, we should assume

that the real cost of electric bills is either $100 or $150, depending on the

model

Equiv annual real cost to own Econo-cool = $ 79.14

plus $150 (real operating cost) = 150.00

$229.14

Equiv annual real cost to own Luxury Air = $ 93.72

plus $100 (real operating cost) = 100.00

$193.72Luxury Air is still more cost effective

37 The equivalent annual cost of the new machine is the 4-year annuity with present value equal to $20,000 This is $7005 This can be interpreted as the extra yearly charge that should be attributed to the purchase of the new machine spread over its life It does not yet pay to replace the equipment since the incremental cash flow provided by the new machine, $10,000 – $5000 = $5000, is less than the equivalentannual cost of the machine.

38 a The equivalent annual cost (EAC) of the new machine over its 10-year life is

found by solving

EAC  annuity factor(5%, 10 years) = $20,000

EAC  7.7217 = $20,000

Therefore, EAC = $2590 Together with maintenance costs of $2000 per year,

the equivalent cost of owning and operating is $4590

The old machine costs $5000 a year to operate, and is already paid for (We assume it has no scrap value and therefore no opportunity cost.) The new machine is less costly You should replace

b If r = 10%, the equivalent annual cost of the new machine increases to $3255,

so the equivalent cost of owning and operating it is now $5255, which is higher than that of the old machine Do not replace

Your answer changes because the higher discount rate implies that the

opportunity cost of the money tied up in the forklift also is higher

39 For the fourth quarter of 2007 business investment in Machinery and Equipment (M&E)increased by 3.4% to $123.7 billion For all of 2007, business investment in M&E rose

by 8.3%

For example, the capacity utilization rate for the fourth quarter of 2007 in the food industry was 79 %, paper was 86.6 % and machinery was 82.2 % The capacity

utilization rate can be an indicator of the likelihood of future capital spending If

the capacity utilization rate is close to 100 %, then it is highly likely that capital

spending will increase in order to further increase capacity The information was

compiled from ”The Daily”, Economic indicators, Summary tables published by

Statistics Canada This information is available through the Statistics Canada

website by using the following web link:

http://www.statcan.ca/english/dai-quo/econind.htm

40 a Present Value =

PV = = $100,000

NPV = –$80,000 + $100,000 = $20,000

b Recall that the IRR is the discount rate that makes NPV equal to zero:

– Investment + PV of cash flows when discounted at IRR = 0

the discount rate, it pays to wait: the value of the tree is increasing faster than the

discount rate When the tree is older and the growth rate is less than r, cutting

immediately is better, since the revenue from the tree can be invested to earn a rate

of r, which is better than the tree is providing

42 a Time Cash flow

0  5

1 30

2 28

The following graph shows a plot of NPV as a function of the discount rate

NPV = 0 when r equals (approximately) either 15.61% or 384% These are

the two IRRs

b Discount rate NPV Develop?

EFFECTIVE ANNUAL COST

Ultra Fast = $55,707.49/PV factor =$55,707.49 /3.605 = $15,452.84Medium Fast= $68,808.13/PV factor =$68,808.13/3.037 = $22,656.61Recommendation – buy Ultra Fast which costs less

b) Effective Annual Cost without salvage value

Ultra Fast = 60,814.33/3.605 = $16,869.44

Medium Fast = 72,780.12/3.037 = $ 23,964.48

c) With salvage value:

Ultra Fast’s effective annual cost at the end of 4 years = $53,392.38/3.037 =

$17,580.63 This is lower than Medium Fast (EAC = $22,656.94) Hence, we should purchase Ultra Fast which will be replaced by Hyper 3MM in 4 years Without salvage value: