chapter 9: net present value and other investment...
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Chapter 9: Net Present Value andOther Investment Criteria
Faculty of Business AdministrationLakehead University
Spring 2003
May 20, 2003
Outline
9.1 Net Present Value
9.2 The Payback Rule
9.3 The Average Accounting Return
9.4 The Internal Rate of Return
9.5 The Profitability Index
9.6 The Practice of Capital Budgeting
1
9.1 Net Present Value
Thenet present value (NPV)of a project measures the difference
between the present value of the project’s future cash flows and
the present value of its costs.
The process of valuing an investment by discounting its future
cash flows is often calleddiscounted cash flow (DCF) valuation.
2
9.1 Net Present Value: An Example
Consider a project with an initial cost of $30 and subsequent
costs of $14 per year. That is, some equipment is purchased at
time 0 for $30 and will cost $14 per year to operate.
The project is expected to generate a cash flow of $24 per year
for eight years.
At the end of the eighth year, the equipment used in the project
will be sold for $2 (salvage value).
What is the net present value of this project?
3
9.1 Net Present Value: An Example
Timing of Cash Flows
0 1 2 3 4 5 6 7 8Year
Initial cost
Inflows
Outflows
Salvage
Net cash flow
-30
-30
24
-14
10
24
-14
10
24
-14
10
24
-14
10
24
-14
10
24
-14
10
24
-14
10
24
-14
2
12
4
9.1 Net Present Value: An Example
This project’s cash flows can be divided in three parts:
1. An outlay of $30 at time 0;
2. An annuity of $10 per year for eight year, the first payment
taking place in year 1;
3. A lump-sum payment of $2 at the end of year 8.
5
9.1 Net Present Value: An Example
Using a discount rate of 12%, the net present value of this project
is
NPV = −30 +10.12
(1−
(1
1.12
)8)
+2
(1.12)8 = $20.48.
The NPV Rule: An investment should be accepted if its net
present value is positive and should be rejected otherwise.
6
9.2 The Payback Rule
Thepayback periodof a project is the time it takes to recover the
project’s initial cost.
The payback rule does not consider the time value of money.
In the previous example, the payback period is exactly 3 years.
7
9.2 The Payback Rule
What is the payback period for each of these projects?
0 1 2 3 4 5 6 7 8Year
Project 1
Project 2
-50 12 12 12 12 12 12 12 12
-50 20 15 10 6 4 3 2 2
8
9.2 The Payback Rule
Let CCFi ≡ cumulative cash flow from projecti.
0 1 2 3 4 5 6 7 8Year
Project 1
CCF1
Project 2
CCF2
-50 12 12 12 12 12 12 12 12-50 -38 -26 -14 -2 10 22 34 46
-50 20 15 10 6 4 3 2 2-50 -30 -15 -5 1 5 8 10 12
9
9.2 The Payback Rule
Project 1’s cost is paid back between year 4 and year 5. The
exact time can be approximated as follows:
Payback period= 4 +2
10− (−2)= 4 +
212
= 4.17years.
Project 2’s cost is paid back between year 3 and year 4. The
exact time can be approximated as follows:
Payback period= 3 +5
1− (−5)= 3 +
56
= 3.83years.
10
9.2 The Payback Rule
The Payback Rule:Accept any investment with a payback
period below some prespecified number of years.
Project 2 pays itself back before project 1. Is project 2 better than
project 1?
11
9.2 The Payback Rule
Let the discount rate be 12%. Then
NPV of project 1 = $9.61.
NPV of project 2 = −$3.75.
That is, if the payback rule were “Only accept projects with a
payback period under 4 years”, then project 2 would be accepted
and project 1 would be rejected, even though
NPV of project 2< 0 < NPV of project 1.
12
9.2 The Payback Rule
Advantages of the Payback Rule
1. Easy to understand.
2. Adjusts for uncertainty of later cash flows.
3. Biased toward liquidity.
13
9.2 The Payback Rule
Disadvantages of the Payback Rule
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.
14
The Discounted Payback Rule
Thediscounted payback periodof a project is the time it takes to
repay the project’s initial cost with thediscountedfuture cash
flows.
This rule thus takes into account the time value of money.
The rule is that a project is accepted if its discounted payback
period is below some prespecified number of years.
15
9.2 The Payback Rule
Supposer = 12%, and let CDCFi ≡ cumulative discounted cash
flow from projecti. Then
0 1 2 3 4 5 6 7 8Year
Project 1
CCF1
CDCF1
Project 2
CCF2
CDCF2
-50 12 12 12 12 12 12 12 12
-50 -38 -26 -14 -2 10 22 34 46-50 -39.3 -29.7 -21.2 -13.6 -6.7 -0.7 4.8 9.6
-50 20 15 10 6 4 3 2 2
-50 -30 -15 -5 1 5 8 10 12-50 -32.1 -20.2 -13.1 -9.3 -7.0 -5.5 -4.6 -3.8
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The Discounted Payback Rule
The discounted payback period for project 1 is between year 6
and year 7, whereas project 2 never pays back its initial cost with
its discounted cash flows.
The discounted payback rule never selects projects with negative
net present value.
17
The Discounted Payback Rule
Advantages of the Discounted Payback Rule
1. Takes the time value of money into account.
2. Easy to understand.
3. Does not accept projects with negative NPV.
4. Biased toward liquidity.
18
The Discounted Payback Rule
Disadvantages of the Discounted Payback Rule
1. May reject projects with positive NPV.
2. requires an arbitrary cutoff point.
3. Ignores cash flows beyond the cutoff date.
4. Biased against long-term projects.
19
9.3 The Average Accounting Return
Theaverage accounting return (AAR)is measured as follows:
Some measure of average accounting profitSome measure of average accounting value
.
We could use, for instance,
Average net incomeAverage book value of investment
.
20
9.3 The Average Accounting Return
Consider a project that requires an initial outlay of $500.
The project has a 5-year life, during which the initial investment
depreciates linearly (straight-line depreciation) to zero.
That is, the initial investment depreciates by $100 each year, andthus its average book value is
500+400+300+200+100+06
=100(5+4+3+2+1)
6
=100× 5×6
2
6=
5002
= 250.
21
9.3 The Average Accounting Return
More specifically, note that
1 + 2 + 3 + . . . + (n−1) + n =n(n+1)
2.
22
9.3 The Average Accounting Return
If an asset of valueA depreciates to 0 overn years, this meansthat it losesA
n of its value each year, and thus its average bookvalue (ABV) is
ABV =A+
(A− A
n
)+
(A− 2A
n
)+ . . .+ 2A
n + An +0
n+1
=An ×n+ A
n × (n−1)+ An × (n−2)+ . . .+ A
n ×2+ An ×1+0
n+1
=A
n(n+1)× (n+(n−1)+(n−2)+ . . .+2+1)
=A
n(n+1)× n(n+1)
2
=A2
.
23
9.3 The Average Accounting Return
Back to our example, suppose the project is expected to generate
the following net incomes:
0 1 2 3 4 5Year
Net income 100 80 70 90 110
The average net income is then
100+80+70+90+1105
= 90.
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9.3 The Average Accounting Return
The average accounting return in our example is then
AAR =90250
= 0.36.
The Average Accounting Return Rule:A project is acceptable
if its AAR is above some target AAR.
25
9.3 The Average Accounting Return
Advantages of the ARR Rule
1. Easy to calculate.
2. Needed information usually available.
26
9.3 The Average Accounting Return
Disadvantages of the ARR Rule
1. Ignores the time value of money.
2. Uses an arbitrary cutoff rate.
3. Based on book values instead of cash flows or market values.
27
9.4 The Internal Rate of Return
Theinternal rate of return (IRR)is the most important alternative
to NPV.
It provides the discount rate at which an investment has a zero
net present value.
The IRR Rule: If the IRR of an investment is greater than the
required rate of return, then the investment is worth undertaking.
28
9.4 The Internal Rate of Return: An Example
0 1 2 3 4 5Year
Cash flow -50 11 13 15 15 14
The internal rate of return in this example is 10.71%:
−50 +11
1.1071+
13(1.1071)2 +
15(1.1071)3 +
15(1.1071)4 +
14(1.1071)5 ≈ 0.
29
9.4 The Internal Rate of Return
With an IRR of 10.71%, will the project be undertaken?
If the firm requires a return of 12%, say, or higher on any of its
projects, then this one won’t be undertaken.
If, on the other hand, the firm requires a return of 9% or higher
on any of its projects, then this one will be undertaken.
30
9.4 The Internal Rate of Return
• There exists a positive IRR to any project with a positive
NPV.
• A project with a negative NPV, i.e. a project that never pays
back the initial investment, has a negative IRR.
• Given a certain discount rate, the fact that projectA, say, has
a greater NPV than projectB does not imply thatA’s IRR is
greater thanB’s IRR, and vice versa.
• A single project may have more than one IRR.
31
Problems with the IRR
Multiple IRRs
Consider the following stream of cash flows:
0 1 2Year
Cash flow -60 155 -100
This project would have two IRRs: 25% and 33.33%.
The maximum number of IRRs a project can have is equal to the
number of times cash flows change sign.
32
Problems with the IRR
Mutually Exclusive Investments
Is the IRR the right rule to use when a firm has access to
mutually exclusive projects?
Consider the two following projects:
0 1 2 3 4 5 6 7 8Year
ProjectA
ProjectB
-50 8 10 11 12 14 15 15 17
-50 16 13 12 12 12 11 10 8
33
Problems with the IRR
Mutually Exclusive Investments
From these cash flows, we find:
IRRA = 16.71% and IRRB = 18.69%.
Is projectB better than projectA?
The NPV ofB is not always greater than that ofA.
34
Problems with the IRR
Mutually Exclusive Investments
Rate NPVA NPVB
5% 30.39 27.40
7% 23.56 22.03
9% 17.53 17.23
11% 12.20 12.94
13% 7.47 9.07
15% 3.25 5.59
35
Problems with the IRR
Mutually Exclusive Investments
ProjectA and projectB have the same NPV when the discount
rate is around 9.5458%. This is the crossover rate.
The Crossover rate is the internal rate of return ofA−B.
Let rc denote the crossover rate. Then, whenr = rc,
NPVA = NPVB ⇒ NPVA − NPVB = 0.
36
Problems with the IRR
Mutually Exclusive Investments
Let CFA,t and CFB,t denote the cash flow at timet of projectA
and projectB, respectively. Then
NPVA − NPVB =8
∑t=0
CFA,t
(1+ r)t −8
∑t=0
CFB,t
(1+ r)t
=8
∑t=0
CFA,t −CFB,t
(1+ r)t
= NPVA−B.
37
Problems with the IRR
Mutually Exclusive Investments
0 1 2 3 4 5 6 7 8Year
ProjectA
ProjectB
A−B
-50 8 10 11 12 14 15 15 17
-50 16 13 12 12 12 11 10 8
0 -8 -3 -1 0 2 4 5 9
38
Problems with the IRR
Mutually Exclusive Investments
Whenr < 9.5458%, then NPVA > NPVB and thus the IRR rule
may contradict the NPV rule.
There is no conflict when Whenr > 9.5458%.
Looking at different streams of cash flows, can you tell whether
the IRR and the NPV rules contradict each other?
39
The IRR Rule
Advantages
• Closely related to the NPV rule.
• Easy to understand and communicate.
Disadvantages
• May provide multiple answers.
• May provide the wrong answer when comparing mutually
exclusive projects.
40
9.5 The Profitability Index
Theprofitability index (PI)is a benefit/cost ratio.
If a project costs $25 and the present value of its future cash
flows is $37.5, then this project has a profitability index of
37.525
= 1.5.
A project with a positive NPV will have a PI greater than one.
A project with a negative NPV will have a PI smaller than one.
41
9.5 The Profitability Index
Mutually Exclusive Investments
This measure may also lead to the wrong decision when selecting
among mutually exclusive projects.
Suppose projectA costs $5 and pays off $11 in one year.
Suppose projectB costs $100 and pays off $121 in one year.
If the discount rate is 10%, then
PIA = 2 > 1.1 = PIB but NPVA = 5 < 10 = NPVB.
42
9.5 The Profitability Index
Advantages
• Closely related to the NPV rule.
• Easy to understand and communicate.
• Useful when capital available is limited.
Disadvantages
• May lead to incorrect decisions when comparing mutually
exclusive projects.
43
9.6 The Practice of Capital Budgeting
Since the NPV tells us what we want to know, why do these
other measures exist?
The NPV is only an approximation. The actual cash flows may
be really different from what is expected. Firms usually use
multiple criteria to evaluate a proposal.
For instance, a positive NPV, a short payback period and a high
AAR mean that the project is probably a good one.
If, on the other hand, the firm receives conflicting signals
(positive NPV, long payback period and low AAR), then it must
be more careful when making its decision.
44
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