Investment Decisions and
Capital Budgeting
Global Financial Management
Campbell R. Harvey
Fuqua School of Business
Duke University
charvey@mail.duke.edu
http://www.duke.edu/~charvey
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Overview
Capital Budgeting Techniques
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Net Present Value (NPV)
» Criterion for capital budgeting
decisions
Special cases:
» Repeated projects
» Optimal replacement rules
Alternative criteria
» Internal Rates of Return (IRR)
» Payback period
» Profitability Index
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Net Present Value
1) Identify base case and alternative
2) Identify all incremental cash flows (Be comprehensive!)
3) Where uncertain use expected values
» Don’t bias your expectations to be “conservative”
4) Discount cash flow and sum to find net present value (NPV)
5) If NPV > 0, go ahead
6) Sensitivity Analysis
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NPV - The Two-Period Case
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Suppose you have a project which has:
» An investment outlay of $100 in 1997 (period 0)
» A safe return of $110 in 1998 (period 1)
» Should you take it?
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What is your alternative?
» Put your money into a bank account at 6%, receive $106
» Gain 4$ in terms of 1998 money
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The project has a positive value!
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Formal Analysis - The Idea
Denote the 1997 and 1998 cash flows as follows:
CF0 = - 100
Cash outflow in period 0
CF1 = 110
Cash return in period 1
Your comparison is a rate of return r of 6% or r=0.06. You invest
only if:
CF0 (1  r )  CF1  0

CF1
CF0 
 NPV  0 
1 r
- 100 * 106
. + 110  0
110
-100 +
 38
.
1.06
The NPV expresses the gain from the investment in 1998 dollars.
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Calculating NPVs
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You have incremental cash flows:
CF0 , CF1 , CF2 , ... , CFT
NPV in year 0 is:
NPV  CF0 

T

t 0
CF1
CF2
CFT


....

(1  r )
(1  r )2
(1  r )T
CFt
(1  r )
t
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Computing NPVs
Example
Step 1:
Year
1997 1998 1999 2000
CF
-100
-50
30
200
Step 2: Determine the PVs of cash flows:
DF
1.000 0.909 0.826 0.751 Total
DCF
-100.0 -45.5
24.8 150.3 = 29.6
Step 3: Sum!
-100.00 - 45.5 + 24.8 + 150.3 = 29.6
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Why Use the NPV Rule?
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We showed that a project with a cash flow:
-100
-50
30
200
had an NPV of 29.6 @ 10%. So what?
Suppose the only shareholder has a bank account where she
can borrow or deposit at 10%.
Take on the project, draw out 29.6 and spend:
Year
Project Cash Flow
Loan Cash Flow
Interest
Balance of account
Payment to shareholder
1997
-100.00
129.60
0.00
-129.60
29.60
1998
-50.00
50.00
12.96
-192.56
0.00
1999
30.00
-30.00
19.26
-181.82
0.00
2000
200.00
-200.00
18.18
0.00
0.00
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What if NPV is negative?
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Suppose you accept a negative NPV project:
Year
Project Cash Flow
Loan Cash Flow
Interest
Balance of account
Payment to shareholder
1997
-100.00
92.04
0.00
-92.04
-7.96
1998
-50.00
50.00
9.20
-151.24
0.00
1999
30.00
-30.00
15.12
-136.36
0.00
2000
150.00
-150.00
13.64
0.00
0.00
» Negative NPV means that you have to spend money today
to be able to undertake the project!
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Replicate the Project with Bonds
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Recall argument about zero coupon bonds
Replicate project with 3 bonds:
» Invest in a 1-year bond with face value 50
» Sell a 2 year bond with face value 30
» Sell a 3 year bond with face value 200
» Include project in your “portfolio”
Year
Project Cash Flow
Bond 1 (1 Year)
Bond 2 (2 Year)
Bond 3 (3 Year)
Portfolio
1997
-100.00
-45.45
24.79
150.26
29.60
1998
-50.00
50.00
1999
30.00
2000
200.00
-30.00
0.00
0.00
» Portfolio has zero cash flows in the future (perfect
replication)
* Value today = NPV!
-200.00
0.00
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Net Present Value (NPV)
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The NPV measures the amount by which the value of the firm’s stock
will increase if the project is accepted.
NPV Rule:
» Accept all projects for which NPV > 0.
» Reject all projects for which NPV < 0.
» For mutually exclusive projects, choose the project with the highest
NPV.
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NPV Example
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Consider a drug company with the opportunity to invest $100
million in the development of a new drug.
» expected to generate $20 million in after-tax cash flows for
the next 15 years.
» the required return is 10%
– What is the NPV of this investment project?
– What if the required return is 20%?
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NPV Example (cont.)
rp = 10%
$20[1  1 / (110
. ) 15 ]
NPV 
 $100
.10
NPV  $52.12 million
rp = 20%
$20[1  1 / (1.20) 15 ]
NPV 
 $100
.20
NPV  $6.49 million
What do you conclude?
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Special Topics: Comparing
Projects with Different Lives
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Your firm must decide which of
two machines it should use to
produce its output.
Machine A costs $100,000, has a
useful life of 4 years, and
generates after-tax cash flows of
$40,000 per year.
Machine B costs $65,000, has a
useful life of 3 years, and
generates after-tax cash flows of
$35,000 per year.
The machine is needed
indefinitely and the discount rate
is rp = 10%.
Year
0
1
2
3
4
5
6
7
8
9
10
…
Machine A Machine B
-65
-100
35
40
35
40
-30
40
35
-60
35
40
-30
40
35
40
35
-60
-30
40
35
40
…
…
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Comparing Projects with Different Lives
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Step 1: Calculate the NPV
for each project.
» NPVA=$26,795
» NPVB=$22,040
» The NPV of A is
received every 4 years
» The NPV of B is
received every 3 years
Year
0
1
2
3
4
5
6
7
8
9
10
…
Machine A Machine B
26795
22040
0
0
0
0
0
22040
26795
0
0
0
0
22040
0
0
26795
0
0
22040
0
0
…
…
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Comparing Projects with Different Lives
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Step 2: Convert the NPVs
for each project into an
equivalent annual annuity.
EAA 
EAB 
$26,795
1  1 / 110
. 

01
.

4



$22,040
3
1  1 / 110
.  


01
.


 $8,453
 $8,863
Year
0
1
2
3
4
5
6
7
8
9
10
…
Machine A Machine B
0
0
8863
8453
8863
8453
8863
8453
8863
8453
8863
8453
8863
8453
8863
8453
8863
8453
8863
8453
8863
8453
…
…
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Comparing Projects with Different Lives
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The firm is indifferent between the project and the equivalent annual
annuity.
Since the project is rolled over forever, the equivalent annual annuity
lasts forever.
The project with the highest equivalent annual annuity offers the
highest aggregate NPV over time.
» Aggregate NPVA = $8,453/.10 = $84,530
» Aggregate NPVB = $8,863/.10 = $88,630
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Special Topics: Replacing an
Old Machine
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The cost of the new machine is $20,000 (including delivery and
installation costs) and its economic useful life is 3 years.
The existing machine will last at most 2 more years.
The annual after-tax cash flows from each machine are given in the
following table.
The discount rate is rp = 10%.
Annual After-Tax Cash Flows
Machine
Year 1
Year 2
Old
$8,000
$6,000
New
$18,000
$15,000
Year 3
$10,000
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Replacing an Old Machine
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Step 1: Calculate the NPV of the new machine.
NPVNew 
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Step 2: Convert the NPV for the new machine into an equivalent
annual annuity.
EANew 
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$18,000 $15,000 $10,000


 $20,000  $16,273
2
3
110
.
(110
. )
(110
. )
$16,273
 $6,544
3
[1  1 / (110
. ) ]


.
10


The NPV of the new machine is equivalent to receiving $6,544
per year for 3 years.
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Replacing an Old Machine (2)
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Step 3: Decide to reinvest machine if EANew>CFOld:
Old
8000
6000
0
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New
6544
6544
6544
Operate the old machine as long as its after-tax cash flows are
greater than EANew = $6,544.
Old machine should be replaced after one more year of
operation.
How did we know that the new machine itself would not be
replaced early?
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Eurotunnel NPV
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One of the largest commercial investment project’s in recent years is
Eurotunnel’s construction of the Channel Tunnel linking France with
the U.K.
The cash flows on the following page are based on the forecasts of
construction costs and revenues that the company provided to
investors in 1986.
Given the risk of the project, we assume a 13% discount rate.
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Eurotunnel’s NPV
Year
Cash Flow
PV (k=13%)
Year
Cash Flow
PV (k=13%)
1986
-GBP457
-457
1999
636
130
1987
-476
-421
2000
594
107
1988
-497
-389
2001
689
110
1989
-522
-362
2002
729
103
1990
-551
-338
2003
796
100
1991
-584
-317
2004
859
95
1992
-619
-297
2005
923
90
1993
211
90
2006
983
86
1994
489
184
2007
1,050
81
1995
455
152
2008
1,113
76
1996
502
148
2009
1,177
71
1997
530
138
2010
17,781
946
1998
544
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NPV
GBP251
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Alternatives to NPV
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Internal Rate of Return (IRR)
Payback
Profitability Index
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Internal Rate of Return
Method
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Calculate the discount rate which makes the NPV zero
» Question: How high could the cost of capital be, so that the
NPV of a project is still positive?
The higher the IRR the better the project
Advantages
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Calculation does not demand knowledge of the cost of capital
Many people find it a more intuitive measure than NPV
Usually gives the same signal as NPV
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Internal Rate of Return (IRR)
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The IRR is the discount rate, IRR, that makes NPV = 0.
T
CFt
NPV  
I 0
t
t 1  1  IRR 
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IRR Rule for investment projects:
» Accept project if IRR > rp.
» Reject project if IRR < rp.
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IRR Example
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Consider, once again, the drug company that has the
opportunity to invest $100 million in the development of a new
drug that will generate after-tax cash flows of $20 million per
year for the next 15 years. What is the IRR of this investment?
The IRR makes NPV = 0.
1  (1  IRR) 15
NPV 
20  100  0
IRR
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This gives IRR = 18.4%.
Accept the project if rp < 18.4%.
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IRR Example (2)
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Consider again the example above
Time
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0
-100.00
1
-50.00
2
30.00
3
200.00
Then the IRR solves:
NPV  100 
50
30
200


0
2
3
1  IRR 1  IRR
1  IRR
» IRR=18.29%
» Accept project if rp<18.29%
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IRR Problems I:
Borrowing or Lending?
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Consider the following two investment projects faced by a firm
with rp = 10%.
Project
B
C
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0
-5000
5000
1
0
2
9800
-9800
IRR
40%
40%
Both projects have an IRR = 40%, but only project A is
acceptable.
» What is happening here?
» How can you modify the IRR rule so that it works?
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NPV Profiles
5000
4000
3000
2000
B
70%
60%
50%
40%
30%
20%
-1000
10%
0
0%
NPV
1000
C
-2000
-3000
-4000
-5000
Discount Rate
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IRR Problems II:
Multiple IRRs
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Consider a firm with the following investment project and a discount
rate of rp = 25%.
Project
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0
-5000
1
16000
2
IRR
-12000 100%, 20%
NPV @ 10% NPV @ 20%
-372
0
Typical if investment at the end:
» Repair environmental damage
» Dismantling of machine
– Nuclear power plants
This project has two IRRs: one above rp and the other below rp. Which
should be compared to rp?
» Should the firm take this project?
– NPV@25%=120
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NPV Profile
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400
NPV
200
0
0%
-200
20%
40%
60%
80%
100%
-400
-600
-800
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-1000
Discount rate
General rule:
IRR works only if sign of CFs
changes once:
» If negative first, then
investment, positive NPV:
IRR>Cutoff
» If positive first, then
financing, positive NPV:
IRR<Cutoff
If pattern changes signs n
times, there will be n different
IRRs!
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IRR Problems III:
Mutually Exclusive Projects with different time horizon
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Consider the following two mutually exclusive projects. The discount
rate is rp = 20%.
Project
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0
1
2
IRR
A
-5,000
8,000
0
B
-5,000
0
9,800
NPV
(k=20%)
60%
1,667
40%
1,806
Despite having a higher IRR, project A is less valuable than project B.
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NPV Profiles
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5000
4000
Project A
Project B
3000
NPV
2000
1000
0
-1000 0
-2000
0.2
0.4
0.6
0.8
Discount Rate, k
-3000
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IRR does not take into
account:
» Capital outlay: project
with higher IRR has lower
NPV (scale effect)
» Time horizon:
– Project A achieves
1
higher return over 1
period
– Project B achieves
mediocre return over 2
periods
Implicit reinvestment
assumption
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IRR Problems IV:
Mutually Exclusive Projects with different scale
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Consider the following two mutually exclusive projects:
Project
A
D
0
-5000
-10000
1
8000
15000
2
0
0
IRR
60%
50%
NPV @ 10% NPV @ 20%
2273
1667
3636
2500
» Project A has higher IRR
» Project D has higher NPV at discount rates of 10% or 20%
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NPV Profiles
5000
A
4000
D
3000
NPV
2000
1000
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
-1000
0%
0
-2000
-3000
Discount Rate
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Payback
Method
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Calculate the time for cumulative cash flows to become positive
The shorter the payback the better
Advantages
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Does not demand input cost of capital
Don’t need to be able to multiply
Gives a feel for time at risk
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Drawbacks
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Arbitrary Ranking. The following projects:
(A) -100
(B) -100
(C) -100
+90
+10
+10
+10
0
+90
0
+90 +100
0
0
+200
all look equally good
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Better ways of coping with risk
» if worried about eg confiscation, adjust cash flows (makes
you think about consequences)
» if worried about risk, use higher discount factor
» recognize time profile of risks
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Not additive, hence combining projects gives different results.
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Payback Example
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Consider the following two investment projects. Assume that rp =
20%.
Project
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0
1
2
A
-1,000
200
800
B
-1,000
200
200
3
Payback
NPV
(k=20%)
300 2.0 yrs.
-104
2,000 2.3 yrs.
463
Which project is accepted if the payback period criteria is 2 years?
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Payback and Money at Risk
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Payback realizes that for duration of project, money is at risk
» More distant cash flows less certain
NPV approach to “Money at Risk”:
Discount rate = Risk free rate + Risk Premium
Example:
Risk free rate = 10%
Risk premium = 5%
Discount factor\Period
@ 10%
@ 15%
Difference
% Difference
1
0.91
0.87
3.95
4.35%
2
0.83
0.76
7.03
8.51%
3
0.75
0.66
9.38
12.48%
4
0.68
0.57
11.13
16.29%
» Much better than payback period!
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Problems with Payback
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Ignores the Time Value of Money
Ignores Cash Flows Beyond the Payback Period
Ignores the Scale of the Investment
Decision Criteria is Arbitrary
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Profitability Index
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Profitability Index
NPV
PI 
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Used when the firm (or division) has a limited amount of capital
to invest.
Rank projects based upon their PIs. Invest in the projects with
the highest PIs until all capital is exhausted (provided PI > 1).
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Profitability Index Example
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Suppose your division has been given a capital budget of $6,000.
Which projects do you choose?
Project
I
NPV
PI
A
1,000
600
0.6
B
4,000
2,000
0.5
C
6,000
2,400
0.4
D
3,000
600
0.2
E
5,000
500
0.1
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Profitability Index Example
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Suppose your budget increases to $7,000.
Choosing projects in descending order of PIs no longer
maximizes the aggregate NPV.
Projects A and C provide the highest aggregate NPV = $3,000
and stay within budget.
Linear programming techniques can be used to solve large
capital allocation problems.
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Conclusions
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NPV has strong attractions:
» based on cash flows - so does not depend on accounting
conventions
» fully reflects time value of money
» takes into account riskiness of project
» gives clear go/no go answer
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