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Chapter 8
Net Present Value and Other Investment Criteria
Net Present Value
- an illustration
Suppose that you are in the real estate business. You are considering construction
of an office block. The land would cost $50,000, and construction would cost a
further $300,000. You foresee a shortage of office space and predict that a year
from now you will be able to sell the building for $400,000 for sure.
The office building is not the only way to obtain $400,000 a year from now. You
could invest in 1-year U.S. Treasury notes. Suppose Treasury notes offer interest
of 7%.
-300,000+(-50,000) =-$350,000
$400,000 a sure cash flow
1 0 r=?
PV = ?
-300,000+(-50,000) =-$350,000
$400,000 a sure cash flow
1 0
Investment in T-Notes
(= -PV = ?)
$400,000 a sure cash flow
0 1 rate of return = 7%
Investment in the
office development
Required rate of return
r = ?
comparable risk (here, riskfree)
Investment in the
U.S. treasury
Required rate of return 7%
Corp2021F_Ch8n.doc
2
Back to the problem:
832,373$9346.0000,40007.1
1000,400 PV
$23,832$350,000-$373,832 cost initial PVNPV
- a comment on risk and present value
look again at the previous example
Suppose you believe the office development is as risky as an investment in the
stock market and that you forecast a 12% rate of return for stock market
investments
12% would be the appropriate opportunity cost of capital
-300,000+(-50,000) =-$350,000 (initial cost)
$400,000 a sure cash flow
1 0 r=7%
Investment in the
office development
Required rate of return
rrisky = ?
comparable risk (here, risky)
Investment in the
U.S. stock market
Required rate of return
12%
PV = ? $400,000 (an expected cash flow)
0 1 %7
??
risky
risky
r
r
Corp2021F_Ch8n.doc
3
143,357$8929.0000,40012.1
1000,400 PV
$7,143$350,000-$357,143 cost initial PVNPV
NPV of a project
We discount the expected future payoff by the rate of return offered by
comparable investment alternatives
※ the discount rate is often known as the opportunity cost of capital
NN
r
CF
r
CF
r
CF
r
CFCFNPV
)1()1()1()1( 33
221
0
0
CF1
1
CF0 CF2 CFN
2 N …
r
Investment in the office development
Required rate of return (opportunity cost of capital)
comparable risk
Investment in the U.S. treasury
Required rate of return r
0
CF1
1
CF0 CF2 CFN
2 N …
-300,000+(-50,000) =-$350,000 (initial cost)
$400,000 Expected cash flow
1 0 r=12%
PV = ?
NPV = PV - initial Cost
Corp2021F_Ch8n.doc
4
- the net present value rule
Managers increase shareholders' wealth by accepting all projects that are
worth more than they cost. Therefore, they should accept all projects with a
positive net present value.
- the meaning of NPV
NPV represent the additional wealth (in terms of today’s money) you can
get if you take the project
a numerical illustration
143,7$12.1
1000,400000,350 NPV
Valuing Long-Lived Projects
- an illustration
Suppose that you are approached by a possible tenant who is prepared to rent your
office block for 3 years at a riskfree fixed annual rent of $25,000. You would need
to expand the reception area and add some other tailor-made features. This would
increase the initial investment to $375,000, but you forecast that after you have
collected the third year's rent the building could be sold for $450,000 for sure
942,57$942,432$000,375$
)07.1(
000,475$
)07.1(
000,25$
07.1
000,25$000,375$
32
NPV
-300,000+(-50,000) =-$350,000 (initial cost)
$400,000 Expected cash flow
1 0 r=12%
PV = ?
NPV = PV - initial Cost
-$375,000 $25,000 $450,000
0 3 r = 7%
$25,000
1 2
$25,000
Corp2021F_Ch8n.doc
5
Using the NPV Rule to Choose among Projects
- Mutually Exclusive Opportunities:
If two investments, X and Y, are mutually exclusive, then taking one of
them means that we cannot take the other.
a vacant lot can be used to do any of the following:
(1) to build an apartment block
(2) to build an office block
(3) to build a gas station
- the NPV rule for mutually exclusive projects
When choosing among mutually exclusive projects, calculate the NPV of each
alternative and choose the highest positive-NPV project.
- Example 8.2: Choosing between two projects
It has been several years since your office last upgraded its office networking
software. Two competing systems have been proposed. Both have an expected
useful life of 3 years, at which point it will be time for another upgrade. One
proposal is for an expensive, cutting-edge system, which will cost $800,000 and
increase firm cash flows by $350,000 a year through increased productivity. The
other proposal is for a cheaper, somewhat slower system. This system would cost
only $700,000 but would increase cash flows by only $300,000 a year. If the cost
of capital is 7%, which is the better option?
5.118$)07.1(
350$
)07.1(
350$
07.1
350$800$
32NPV
3.87$)07.1(
300$
)07.1(
300$
07.1
300$700$
32NPV
Choose the one with higher positive NPV
0 1 2 3
-800 350 350 350
-700 300 300 300
Faster
Slower
7%
Corp2021F_Ch8n.doc
6
The Internal Rate of Return
- calculating the Internal Rate of Return (IRR)
Definition of IRR:
IRR is the discount rate that makes NPV = 0
NNCFCF
CFCFCFNPV
)ratediscount 1()ratediscount 1(
)ratediscount 1()ratediscount 1(
33
221
0
0)1()1()1()1( 3
32
210
NN
IRR
CF
IRR
CF
IRR
CF
IRR
CFCFNPV
※ to get IRR, we do not need to know the (opportunity) cost of capital
a numerical illustration
32 )1(
35
)1(
35
)1(
35250
RRRNPV
14.0250
35
352500
)1(
35
)1(
35
)1(
352500
32
IRR
IRR
IRRIRRIRRNPV
0
CF1
1
CF0 CF2 CFN
2 N …
0 1 2 3
-250 35 35 35 … …
…
0 1 2 3
-250 35 35 35 …
R
…
…
Corp2021F_Ch8n.doc
7
- IRR measures the rate of return of an investment
some important properties of IRR:
(1) IRR is a single rate of return that summarizes the merits of a project
(2) IRR depends only on the cash flows of a particular investment, not on
rates offered elsewhere
an illustration (one-period case)
Consider a project that costs $100 today and pays $110 in one year. Suppose
you were asked, "What is the return on this investment?" What would you
say?
(1) rate of return:
%10100
100110)(
%10
110$)1(100$)(
rB
r
rA
(2) )1(
110$100$
RNPV
%10100
1001101
100
110
)1(
110$100$0
IRR
IRRNPV
0 1
-$100 $110
Corp2021F_Ch8n.doc
8
a demonstration on the relationship between NPV and IRR:
(A) Rate of return:
0
1
0
01 )1(P
Pr
P
PPr
(B) Calculating NPV & IRR:
NPVP
PR
R
PPNPV
0
110 )1(
)1(
Let NPV = 0, we can have
0
01
0
1)1(
P
PPIRR
P
PIRR
- an example from the bond market
NN YTM
F
YTM
CP
YTM
CP
TYM
CPP
)1()1()1()1( 20
Buy the bond:
IRRYTM
YTM
F
YTM
CP
YTM
CP
TYM
CPPNPV
NN
0)1()1()1()1( 2
※ YTM (yield to maturity) is the rate of return by holding the bond
through maturity
1 3 N-1 2
CP
N …
CP CP CP & F CP
Future cash flows of a coupon bond:
r=YTM
1 3 N-1 2
CP
N …
CP CP CP & F CP -P0
0
0 1
0P 1P
Corp2021F_Ch8n.doc
9
- the primary concern of managers: whether the project's return is higher or lower
than the opportunity cost of capital
The internal rate of return (IRR) rule:
Invest in any project offering a rate of return that is higher than the
opportunity cost of capital
The NPV profile
- the relationship between NPVs and discount rates
),()1()1()1()1(
1,1
13
1
32
1
2
1
101 NPVr
r
CF
r
CF
r
CF
r
CFCFNPV
NN
),()1()1()1()1(
22
23
2
32
2
2
2
102 NPVr
r
CF
r
CF
r
CF
r
CFCFNPV
NN
…………
),()1()1()1()1( 3
32
210 KKN
K
N
KKK
K NPVrr
CF
r
CF
r
CF
r
CFCFNPV
0
CF1
1
CF0 CF2 CFN
2 N …
The project provides
IRR
provides r
Other investment
opportunities in the
market with equivalent
risk and maturity the project’s cost
of capital = r
Equivalent risk
Corp2021F_Ch8n.doc
10
The relationship between the NPV rule and the IRR rule
- assuming that we have the following normal case:
The NPV rule: Invest in any project that has a positive NPV when its cash
flows are discounted at the opportunity cost of capital.
The internal rate of return (IRR) rule: Invest in any project offering a rate of
return that is higher than the
opportunity cost of capital
- the logic
NN
rateDiscount
CF
rateDiscount
CF
rateDiscount
CFCFNPV
)1()1()1( 221
0
When NPV=0, the discount rate is called IRR.
NN
IRR
CF
IRR
CF
IRR
CFCF
)1()1()1(0
221
0
NPV
Discount rate 0
The normal case:
a line with a negative slope (between
NPV and the discount rate)
0
CF1
1
CF0 CF2 CFN
2 N …
NPV
Discount rate = IRR
Discount rate
The turning point of NPV
0
Corp2021F_Ch8n.doc
11
(1) when IRR < r (Note: r = cost of capital)
NN
IRR
CF
IRR
CF
IRR
CFCF
)1()1()1(0
221
0
0)1()1()1( 2
210
NN
r
CF
r
CF
r
CFCFNPV
if IRR < r, then reject the project
＊under the normal case:
IRR < cost of capital NPV of the project < 0
(2) when IRR > r (Note: r = cost of capital)
NN
IRR
CF
IRR
CF
IRR
CFCF
)1()1()1(0
221
0
0)1()1()1( 2
210
NN
r
CF
r
CF
r
CFCFNPV
if IRR > r, then accept the project
＊under the normal case:
IRR > cost of capital NPV of the project > 0
NPV
Discount rate = IRR
Discount rate
The turning point of NPV
0
Accept
if the cost of capital falls within this range
NPV
IRR
Discount rate
The turning point of NPV
0
if the cost of capital falls within this range
Reject
Corp2021F_Ch8n.doc
12
(3) a summary
The rate of return rule will give the same answer as the NPV rule as long
as the NPV of a project declines smoothly as the discount rate increases.
- an illustration
32 )1(
000,475$
)1(
000,25$
1
000,25$000,375$0
IRRIRRIRR
000,150$)01(
000,475$
)01(
000,25$
0.1
000,25$000,375$
32
481,206$)50.01(
000,475$
)50.01(
000,25$
50.01
000,25$000,375$
32
the trial and error mechanism
the IRR must lie somewhere between zero and 50%
Figure 8.3
a discount rate of 12.56% gives an NPV of zero
a spreadsheet or specially programmed financial calculator
(1) the opportunity cost of capital < 12.56% NPV > 0
(2) the opportunity cost of capital > 12.56% NPV < 0
NPV
Discount rate = IRR
Discount rate
The turning point of NPV
0
if the cost of capital falls within this range
Reject
if the cost of capital falls within this range
Accept
0 1 2 3
-375,000 25000 25000 475000
Corp2021F_Ch8n.doc
13
Some Pitfalls with the Internal Rate of Return Rule
1. Lending or Borrowing?
- Consider the following projects:
Project C0 C1 IRR (%) NPV at 10%
H -100 +150 +50 +$36.4
I +100 -150 +50 -$36.4
each project has an IRR of 50%
Does this mean that the two projects are equally attractive?
Project H: we are paying out $100 now and getting $150 back at the end of the
year
Project I: we are getting paid $100 now but we have to pay out $150 at the end
of the year
when you lend money, you want a high rate of return; when you borrow,
you want a low rate of return
Note: The NPV rule will give us the correct answer
r
NPV The NPV profile
r
NPV The NPV profile
Corp2021F_Ch8n.doc
14
2. Mutually Exclusive Projects
- an illustration
Think once more about the two office-block proposals from previous sections.
You initially intended to invest $350,000 in the building and then sell it at the end
of the year for $400,000. Under the revised proposal, you planned to invest
$375,000, rent out the offices for 3 years at a fixed annual rent of $25,000, and
then sell the building for $450,000.
C0 C1 C2 C3 IRR (%) NPV at 10%
initial -350,000 +400,000 14.29% +$23,832
Revised -375,000 +25,000 +25,000 +475,000 12.56% +$57,942
The IRR rule: Choose the initial proposal
The NPV rule: Choose the revised proposal
- Figure 8.4
the two NPV profiles cross at an interest rate of 11.72%
(1) opportunity cost of capital > 11.72% the initial proposal is the
superior investment
(2) opportunity cost of capital < 11.72% the revised proposal is the
superior investment
for the 7% cost of capital that we have assumed, the revised proposal is the
better choice
the IRR is simply the discount rate at which NPV equals zero
IRR = 14.29% for the initial proposal
IRR = 12.56% for the revised proposal
the higher IRR for the initial proposal does not mean that it has a higher
NPV
Corp2021F_Ch8n.doc
15
- an example based on self-test 8.4
You are now offered to choose between the following two opportunities:
a. invest $1,000 today and quadruple your money (i.e., a 300% return) in one
year with no risk
b. Invest $1 million for one year at a guaranteed 50% return
Which one will you take? Note that safe securities still yield 7.5%.
Project a:
%300000,1
000,1000,4
aIRR
93.720,2$075.1
000,4000,1 aNPV
Project b:
%50000,000,1
000,000,1000,500,1
bIRR
84.348,395$075.1
000,500,1000,000,1 bNPV
- a caveat to remember:
The goal: to increase your wealth
The question: Which of the following is more consistent with the goal when
choosing between mutually exclusive projects?
(1) Choosing a project with high positive NPV
(2) Choosing a project with higher IRR
0 1
-1,000 4,000
r = 7.5%
0 1
-1,000,000 1,500,000
r = 7.5%
Corp2021F_Ch8n.doc
16
3. Multiple Rates of Return
- an illustration
King Coal Corporation is considering a project to strip-mine coal. The project
requires an investment of $210 million and is expected to produce a cash inflow of
$125 million in the first 2 years, building up to $175 million in years 3 and 4.
However, the company is obliged in year 5 to reclaim the land at a cost of $400
million.
5432 )1(
400$
)1(
175$
)1(
175$
)1(
125$
1
125$210$
rrrrrNPV
54
32
)03.01(
400$
)03.01(
175$
)03.01(
175$
)03.01(
125$
03.01
125$210$0
NPV
54
32
)25.01(
400$
)25.01(
175$
)25.01(
175$
)25.01(
125$
25.01
125$210$0
NPV
- Figure 8.5
the investment has an IRR of both 3% and 25%
Normal cash flows (or conventional cash flows)
All the negative cash flows precede all the positive cash flows.
the sign of the cash flows changes only one time throughout the
life of the project
0 1 5 2 3
-210 125 125 175 -400 175
4
Corp2021F_Ch8n.doc
17
- examples:
- for a stand-alone project with normal cash flows
(1) the project will have one IRR
(2) the IRR rule and the NPV rule generate the same result
- IRR and non-normal cash flow
Suppose we have a strip-mining project that requires a $60 investment. Our cash
flow in the first year will be $155. In the second year, the mine will be depleted,
but we will have to spend $100 to restore the terrain.
2)1(
100
)1(
15560
rrNPV
Discount Rate NPV
0% -$5.00
10% - 1.74
20% - 0.28
25% 0
30% 0.06
32% 0.03
33.33% 0
40% - 0.31
the NPV profile What's the IRR?
IRR = 25.0% and 33.33%
0 1 2 3 4 5
-30
0
240 180 120 60 0 (A)
(B)
(C)
30 -75 75 -56 75
-435 234 180 145 267
390 (D) 256 76 60 -120
100
30
300
0 1 2
-60 155 -100
Corp2021F_Ch8n.doc
18
(reference only) why do we have multiple IRRs?
the cash flows are non-normal
02
21
)1()1()1(0 C
IRR
C
IRR
C
IRR
CNPV
N
N
001
12
21 CyCyCyCyC N
NN
N
- Using the Modified IRR when there are multiple IRRs
The rule:
Combine the cash flows that make the project's cash flows nonnormal with
the closest adjacent previous cash flows until the sum becomes positive
Here the cash flow of "-$400" at Year 5 is the one causing the project's
cash flows nonnormal.
Step #1: Move it to Year 4 (discounted at the opportunity cost of
capital)
158)333(1752.01
4001754,
YearNewCF
5 4
-400 175
-333
discounted at the cost of capital
0 1 5 2 3
-210 125 125 175 175-333
= -158
4
0 1 5 2 3
-210 125 125 175 -400 175
4
NPV The NPV Profile
r
25% 33.33%
Corp2021F_Ch8n.doc
19
Step #2: move the resulting cash flow backward one more year
43$2.01
1581753,
YearNewCF
Alternatively,
43$)2.01(
400
)2.01(
175175
23,
YearNewCF
The Modified IRR:
%22
)1(
43$
)1(
125$
)1(
125$210$0
32
MIRR
MIRRMIRRMIRRNPV
(MIRR = 22%) > (Opportunity cost of capital = 20%)
NPV > 0 (since the resulting cash flow is normal now)
A summary:
4 3
-158 175
-132
discounted at the cost of capital
0 1 5 2 3
-210 125 125 175 -400 175
4
The first step: discount the cash flows at the cost of capital until we have a positive cumulated cash flow
The second step: find the IRR of the normal cash flow
0 1 5 2 3
-210 125 125 43
4
0 1 5 2 3
-210 125 125 175 -132
= +43
4
Corp2021F_Ch8n.doc
20
The Profitability Index
- it measures the net present value of a project per dollar of investment:
also known as the benefit-cost ratio
investment initial
NPVIndexity Profitabil
for the initial proposal to construct an office building
068.0350,000
23,832Indexity Profitabil
- any project with a positive profitability index must also have a positive NPV
00 investmentinitial
NPVNPV
- when to use the profitability index?
there is a limit on the amount the company can spend calculating the
profitability index pick those projects that have the highest profitability
index
- an illustration
Assume that you are faced with the following investment opportunities:
All three projects are attractive, but suppose that the firm is limited to spending
$10 million.
Suggested answer:
Based on the NPV rule:
Choose the highest NPV: C D E
Based on PI:
Choose he highest PI: D E C
Corp2021F_Ch8n.doc
21
- another example (a caveat for using the Profitability Index)
Case #1: Suppose that the fund available for investment is only $240. We have
the following projects to consider.
Project Initial Investment NPV Profitability Index
A 160 400 2.5
B 70 147 2.1
C 80 160 2.0
Case #2: Suppose that the fund available for investment is only $160. We have
the following projects to consider.
Project Initial Investment NPV Profitability Index
A 140 294 2.1
B 70 140 2.0
C 85 161.5 1.9
Capital Rationing
- it refers to a shortage of funds available for investment
I. Soft Rationing
the capital rationing is imposed by top management
senior management may impose a limit on the amount that junior managers
can spend junior mangers set their own priorities
for example: firms with rapid growth could impose soft rationing of capital or
other resources
a tradeoff between:
1. forgoing good projects
2. reducing overinvestment in bad projects
II. Hard Rationing
it means that the firm actually cannot raise the money it needs may be
forced to pass up positive-NPV projects need to select the package of
projects that is within the company's resources and yet gives the highest net
present value this is when the profitability index might be useful
Corp2021F_Ch8n.doc
22
The Payback Rule
- it is the length of time before you recover your initial investment
- the payback rule: a project should be accepted if its payback period is less than a
specified cutoff period
an appropriate cutoff period must be determined
- an example
Payback period = 34286.2175
1251253102
Payback period = 5.28
88202
- an illustration
(1) it may accept a negative NPV project
compare projects F, G, and H:
The cutoff period is 2 years
Accept projects: F, G, and H
Positive NPV: F
Negative NPV: G and H
0 1 5 2 3
-20 8 8 8 8 8
4
the payback period
0 1 2 3
-310 125 125 175 175
4
(-310)+125 =-185
(-185)+125 =-60
Corp2021F_Ch8n.doc
23
(2) it does not consider any cash flows that arrive after the payback period
biased towards short-term projects biased towards liquidity
※ cash flows that arrive after the payback period are ignored a firm
uses the same cutoff regardless of project life tend to accept too
many shortlived projects and reject too many long-lived ones
another example
(3) it gives equal weight to all cash flows arriving before the cutoff period
it ignores the time value of money
Advantages and Disadvantages of the Payback Period Rule
Advantages
1. Easy to understand.
2. Biased towards liquidity.
◎ choose the one with larger cash flows in the early stage of the
project
Disadvantages
1. Ignores the time value of money.
2. Requires an arbitrary cutoff point.
3. Ignores cash flows beyond the cutoff date.
4. Biased against long-term projects, such as research and development,
and new projects.
0 1 5 2 3
-40 15 25 10 120 180
4
the payback period: 2.5 years
cash flows are ignored!
0 1 5 2 3
-20 8 8 8 8 8
4
the payback period
cash flows are ignored!
Corp2021F_Ch8n.doc
24
The Discounted Payback Period Method
- the discounted payback period is the length of time until the sum of the discounted
cash flows is equal to the initial investment
the discounted payback rule would be:
Based on the discounted payback rule, an investment is acceptable if its
discounted payback is less than some prespecified number of years.
- an illustration
Suppose that we require 12.5 percent return on new investments. We have an
investment that costs $300 and has cash flows of $100 per year for five years.
Year Raw CF Cumulated raw CF
Discounted CF Cumulated discounted CF
1 100 100 )125.01(100 89 89
2 100 200 2)125.01(
100 79 168
3 100 300 3)125.01(
100 70 238
4 100 400 4)125.01(
100 62 300
5 100 500 5)125.01(
100 55 355
Payback period = 3100
1001001003103
Discounted payback period = 455
627079893004
(1) the regular payback is exactly three years
(2) the discounted payback is four years
0
1 2 3
100 100
4
(PV0=89)
5
100 100 100
(PV0=79) (PV0=70) (PV0=62) (PV0=55)
-300
the payback the discounted payback
Corp2021F_Ch8n.doc
25
- advantages and disadvantages of the discounted payback rule:
Advantages
1. Includes time value of money.
※ an improvement over the payback period rule
2. Never accepts normal-cash-flow investments whose NPV is negative
※ an improvement over the payback period rule
PVCFNPV 0
method periodpayback d Discounte the
on basedback pay never llproject wi The
0 0
CFPVthenNPVIf
＊ an example
3. Biased towards liquidity.
※ similar to the payback period
0
1 2
CF2 CF1
N
PV
…
CFN CF0
0 1 2 3
-$300 $200 $100 $10
10%
8182.1811.01200
645.822)1.01(
100
513.73)1.01(
10
PV = 181.8182+82.645+7.513 = 271.9762 Discounted payback period = ? It never pays back NPV = -300+181.8182+82.645+7.513 = -$28.0234
Corp2021F_Ch8n.doc
26
Disadvantages
1. May reject positive NPV investments
※ similar to the payback period
Discounted payback period = 2.35 years > the benchmark (i.e., 2 years)
reject the project
2. Requires an arbitrary cutoff point.
※ similar to the payback period
3. Ignores cash flows beyond the cutoff date.
※ similar to the payback period
4. Biased against long-term projects, such as research and development, and
new projects.
※ similar to the payback period
※ this is due to item 3 above
3 2 0 1
$100 $100 $100
4
$100
5
$100
10%
09.62
30.68
13.75
64.82
91.90
5
4
3
2
1
)1.01(
100
)1.01(
100
)1.01(
100
)1.01(
100
)1.01(
100
-$200
NPV = -200+90.91+82.64+75.13+68.30+62.09 = $179.07
Corp2021F_Ch8n.doc
27
Some issues in choosing between mutually exclusive projects
Problem 1: The Investment Timing Decision
- When is it best to commit to a positive-NPV investment?
this is in fact a mutually-exclusive-projects case
- an illustration
You are choosing when to start the project. The cost of the computer is expected
to decline from $50,000 today to $45,000 next year, and so on. The new computer
system is expected to last for 4 years from the time it is installed. The cost of
capital is 10%.
0 2 1
…
initiate the project
NPV
CF1 CF2
CF1
…
(1) 0
(2) 1
(2) 0
(3) 2
(3) 1
…
CF2
initiate the project
NPV
CF1
0 2 1
NPV0=20 PV0(cost)= -50 PV0(saving)=70 NPV0=20
NPV0=25/(1.10)
= 22.7
PV1(cost)= -45 PV1(saving)=70 NPV1=25
PV2(cost)= -40 PV2(saving)=70 NPV2=30 NPV0=30/(1,10)2
=24.8
……...
NPV0=25.5
NPV0=25.3
NPV0=24.2
initiate the project
NPV
Corp2021F_Ch8n.doc
28
For the purchase year of Year 1:
25$4570$)(cos)( 11)(1 tPVsavingPVNPV YearYearpurchasetheYear
7272.22$)10.01(
25$0
YearNPV
For the purchase year of Year 4:
37$3370$)(cos)( 44)(4 tPVsavingPVNPV YearYearpurchasetheYear
2715.25$)10.01(
37$40
YearNPV
Table 8.1: a summary of the result
※ you maximize net present value today by buying the computer in year
3
Problem 2: The Choice between Long and Short-Lived Equipment
Note:
The underlying assumption:
The Mutually exclusive projects will be repeated forward
indefinitely.
- an illustration
Suppose the firm is forced to choose between two machines, I and J. The two
machines are designed differently but have identical capacity and do exactly the
same job.
Machine I costs $15,000 and will last 3 years. It costs $4,000 per year to run.
Machine J is an "economy" model, costing only $10,000, but it will last only 2
years and costs $6,000 per year to run.
Corp2021F_Ch8n.doc
29
Costs (Thousands of dollars; Year) C0 C1 C2 C3 NPV at 6%
Machine I 15 4 4 4 $25,69
Machine J 10 6 6 -- $21.00
0 1 2 3
$4
$6
$4 $4
$6
$15
$10
0 1 5 2 3 4 6
$4 …
$6
$4 $4 $15 $4 $4
$6 $10 $6
$6
$15
$10 $6 $10
…
Corp2021F_Ch8n.doc
30
Convert to Equivalent Annuity:
Should we take machine J, the one with the lower present value of costs?
Is the annual cost of using J lower than that of I?
Calculate the equivalent annual annuity cost:
Costs (Thousands of dollars; Year) C0 C1 C2 C3 NPV at 6%
Machine I 15 4 4 4 $25.69
Equivalent Annual Annuity 9.61 9.61 9.61 $25.69
Equivalent annual annuity × 3-year annuity factor
= PV of costs = $25,690
610,9$
6730.2)3%,6(690,25
C
CPVIFAC
0 1 2 3
EAI
EAJ
EAI EAI
EAJ
0 1 5 2 3 4 6
EAI …
EAJ
EAI EAI EAI EAI
EAJ EAJ EAJ EAJ …
Corp2021F_Ch8n.doc
31
Costs (Thousands of dollars; Year) C0 C1 C2 NPV at 6%
Machine J 10 6 6 $21.00
Equivalent Annual Annuity 11.45 11.45 21.00
450,11$
6730.2)3%,6(000,21
C
CPVIFAC
A summary plot:
- a rule for comparing assets with different lives: Select the machine that has the
lowest equivalent annuity.
- Example 8.3: Equivalent Annual Annuity
You need a new car. You can either purchase one outright for $15,000 or lease
one for 7 years for $3,000 a year. If you buy the car, it will be worth $500 to you
in 7 years. The discount rate is 10%. Should you buy or lease? What is the
maximum lease payment you would be willing to pay?
743,14$)07.1(
500000,15
7PV
028,3
)7%,10(743,14
PMT
PVIFAPMT
9.61 9.61 9.61 ….
6% 1 0 2 3
Machine I
Machine J 11.45 11.45 …..
0
r1
=
?
7 2 …
3,000 3,000 … 3,000
10%
3,028 3,028 3,028 ……
0
r1
=
?
7 2 …
15000
3000 3000 … 3000
-500
10%
Corp2021F_Ch8n.doc
32
- Example 8.4: Another Equivalent Annual Annuity
Low-energy lightbulbs typically cost $3.50, have a life of 9 years, and use about
$1.60 of electricity a year. Conventional lightbulbs are cheaper to buy for they
cost only $.50. On the other hand, they last only about a year and use about $6.60
of energy. If the discount rate is 5%, which product is cheaper to use?
Annual cost:
Low-energy bulb: 49.0
1078.7)9%,5(50.3
C
CPVIFAC
1.60 + 0.49 = 2.09
Conventional bulb: 17.760.6)05.01(50.0
0 1 9 2 …
2.09 2.09 … 2.09
7.17
5%
0 1 9 2 …
3.50 1.60 1.60 … 1.60
0.50 6.60
5%
Corp2021F_Ch8n.doc
33
Another presentation:
0 1 9 2 …
2.09 2.09 … 2.09
7.17 7.17
5%
7.17 …
…
…
…
0 1 9 2 …
3.50 1.60 1.60 … 1.60
0.50 6.60
5%
0.50
0.50
6.60
….
6.60
0.50
Corp2021F_Ch8n.doc
34
Problem 3: When to Replace an Old Machine
- an illustration
You are operating an old machine that will last 2 more years before it gives up the
ghost. It costs $12,000 per year to operate. You can replace it now with a new
machine that costs $25,000 but is much more efficient ($8,000 per year in
operating costs) and will last for 5 years. Should you replace the machine now or
stick with it for a while longer? The opportunity cost of capital is 6%.
$58.70 )06.01(
8
)06.01(
8
)06.01(
8
)06.01(
8
)06.01(
825
543
2
NPV
935.13
2124.470.58
)5%,6(70.58
C
C
PVIFAC
0 1 5 2 3
25 8 8 8 8
6%
8
4
12.00 12.00
0 1 5 2 3
12.00 12.00
13.93
6% 4
13.93 13.93 13.93 13.93
Old
New
0 1 5 2 3
25 8 8 8 8
13.93
6%
8
4
13.93 13.93 13.93 13.93 EAA
Corp2021F_Ch8n.doc
35
- Homework: Self-test 8.8 on page 261
Suggested answer:
Machine K:
7355.991,11)10.01(
200,1
)10.01(
100,1000,10
2
NPV
67.909,6$
7355.1)2%,10(7355.991,11
K
KK
EAA
EAAPVIFAEAA
Machine L:
9446.812,14)10.01(
300,1
)10.01(
200,1
)10.01(
100,1000,12
32
NPV
39.956,5$
4869.2)3%,10(9446.812,14
L
KL
EAA
EAAPVIFAEAA
0 1 2 3
12,000 1,100 1,200 1,300
10%
L
EAAL 5,956.39 5,956.39 5,956.39
0 1 2 3
6,909.67 6,909.67
K
10%
EAAK
10,000 1,100 1,200 --
0 1 2 3
12,00
0
1,100 1,200 1,300
K
10%
L
10,000 1,100 1,200 --
Corp2021F_Ch8n.doc
36
a. Choose Machine L
b. Keep using the existing one
0 1 2 3
6,909.67 6,909.67
10%
EAAK
EAAL 5,956.39 5,956.39 5,956.39
0 1 2 3
6,909.67 6,909.67
10%
EAAK
EAAL 5,956.39 5,956.39 5,956.39
Existing 2,500+1,800 =$4,300
Corp2021F_Ch8n.doc
37
A Last Look
- NPV is the only rule that consistently can be used to rank and choose among
mutually exclusive investments
the only instance in which NPV fails as a decision rule occurs when the firm
faces capital rationing
- Table 8.3
Note: the frequency of use of the various capital budgeting methods on a scale
of 0 (never) to 4 (always)
(1) 75% of U.S. and Canadian companies either always or almost always use NPV
or IRR to evaluate projects
(2) payback period is used by about 57% of the managers, far less than NPV and
IRR
(3) profitability index is routinely computed by only about 12% of firms
(4) both the IRR and NPV methods are used more frequently in large firms than in
small firms
(5) the payback period is used more frequently in small firms than in large firms
