Unit 9 Project Investment Criteria
Net Present Rule
The net present value rule: An investment should be accepted if the net present value is positive
and rejected if it is negative. In the unlikely event that the net present value turned out to be
exactly zero, we would be indifferent between taking the investment and not taking it.
Case: Suppose we are asked to decide whether or not a new consumer product should be launched.
Based on the projected sales and costs, we expect that the cash flows over the five year life of the
project will be $2,000 in the first two years, $4,000 in the next two, and $5,000 in the last year. It
will cost $10,000 to begin production. We use a 10 percent discount rate to evaluate new products.
What should we do here?
Solution
Given the cash flows and discount rate, we can calculate the total value of the product by
discounting the cash flows back to the present:
Present Value = 2,000/1.1 + 2,000/1.12 +4,000/1.13 + 4,000/1.14 +5,000/1.15
= 1,818 + 1,653 + 3,005 + 2,732 + 3,105
= $ 12,313
NPV = 12,313 –10, 000 = $2,313. This is positive. Based on the NPV rule, we should take on
the project.
The Payback Rule
How many years do we have to wait until the accumulated cash flows from this investment equal
or exceed the cost of the investment? Suppose the initial investment is $50,000. After the first year,
the firm has recovered $30,000, leaving $20,000. The cash flow in the second year is exactly
$20,000. So this investment “pay for itself” in exactly two years. Put another way, the payback
period is two years. If we require a payback of, say, three years or less, then this investment is
acceptable. This illustrates the payback period rule:
Based on the payback rule, an investment is acceptable if its calculated payback period is less
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than some predetermined number of years.
Case: Calculating Payback
The projected cash flows from a proposed investment are:
Year
Cash Flow
1
$100
2
200
3
500
This project costs $500. What is the payback period for this investment?
Solution
After the first two years, the cash flows total $300. The third year cash flow is $500, so we
have to wait 200/500 = 0.4 year to do this. The payback period is thus 2.4 years, or about two
years and five months.
Advantages of the Payback Period Rule
 Easy to understand
 Advantages for uncertainty of later cash flows
 Biased towards liquidity
Disadvantages of the Payback Period Rule
 Ignore the time value of money
 Requires an arbitrary cutoff point.
 Ignores cash flows beyond the cutoff date.
 Biased against long-term projects, such as research and development, and new project.
The Discounted Payback
The discounted payback rule would be: Base on the discounted payback rule, an investment is
acceptable if its discounted payback is less than some prescribed number of years.
Case: To see how we might calculate the discounted payback period, suppose that we require a
12.5 percent return on new investment. We have an investment that costs $300 and has cash flows
of $100 per year for five years. To get the discounted payback, we have to discount each cash flow
at 12.5 percent and then start adding them. The following table expresses this calculation.
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Cash Flow
Accumulated Cash Flow
Year
Undiscounted
Discounted
Undiscounted
Discounted
1
$100
$89
$100
$89
2
100
79
200
168
3
100
70
300
238
4
100
62
400
300
5
100
55
500
355
Solution
Because it includes the time value of money, the discounted payback is the time it takes to
break even in an economics of financial sense. For undiscounted payback rule, the payback
period is 3 years; and for discounted payback rule, the payback period is 4 years.
Advantages of the Discounted Payback Rules
 Includes time value of money.
 Easy to understand.
 3. Biased towards liquidity
 4.Does not accept negative estimated NPV investments.
Disadvantages of the Discounted Payback Rules
 May reject positive NPV investment.
 Requires an arbitrary cutoff point.
 Ignores Cash flows beyond the cutoff date.
 Biased against long-term projects, such as research and development, and new projects.
The Internal Rate of Return
By definition, IRR is the discount rate that makes the NPV of an investment zero. In other words,
the IRR on an investment is the required return that results in a zero NPV when it is used as the
discount rate. And accordingly, here is the IRR rule:
Based on the IRR rule, an investment is acceptable if the IRR exceeds the required return, it
should be rejected otherwise.
Case: Suppose an investment costs $100, and has a cash flow of $60 per year for two years. How
to find the IRR rate?
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Solution
We have the following equation:
NPV = 0 = —100 + [60/(1 +IRR)] + [1 + 60/(1 + IRR)2]
Compare to NPV, suppose the discount rate is 10 percent, we get the following result:
NPV = —100 + [60/(1 + 1.1)] + [1 + 60/(1 + 1.1)2] = $4.13
Generally speaking, we find the IRR by trial and error, either by hand or by calculator.
Mutually Exclusive Investment
Even if there is a single IRR, another problem can arise concerning mutually exclusive investment
decisions. If two investments, X and Y, are exclusive, then taking one of them means that we
cannot take the other. Given two or more mutually exclusive investments, which one is the best?
The answer is simple enough: the best one with the largest NPV. Can we also say that the best one
has the highest return? As we show, the answer is no.
Case: To illustrate the problem with the IRR rule and mutually exclusive investments, consider the
following cash flows from two mutually exclusive investments:
Year
Investment A ($)
Investment B ($)
0
-100
-100
1
50
20
2
40
40
3
30
50
4
20
60
The IRR for A is 24 percent, and IRR for B is 21 percent. Simple intuition suggests that
Investment A is better because of its higher return. Unfortunately, simple intuition is not always
correct. To see why investment A is not necessarily the better of the two investments, we’ll
calculate the NPV of these investments for different required returns:
Discount rate %
NPV for A ($)
NPV B ($)
0
60.00
70.00
5
43.13
47.88
10
29.06
29.79
15
17.18
14.82
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20
7.06
2.31
25
-1.63
-8.22
If you compare the NPVs, you’ll see that which investment has the higher NPV depends on
our required return. B has greater total cash flow, but it pays back more slowly than A. As a result,
it has a higher NPV at lower discount rates. In our example, the NPV and IRR rankings conflict
for some discount rates. If our required return is 10 percent, for instance, then B has the higher
NPV and is thus the better of the two even though A has the higher return. If our required return is
15 percent, then there is no ranking conflict: A is better.
Calculating the Crossover Rate
However, we are interested in analyze another case to understand the following idea: How can we
determine just what this crossover point is? The crossover rate, by definition, is the discount rate
that makes the NPVs of two projects equal. To illustrate, suppose we have the following two
mutually exclusive investments:
Year
Investment A ($)
Investment B ($)
0
-400
-500
1
250
320
2
280
340
To find the crossover, first consider moving of Investment A and into Investment B. by so
doing, you will have to invest an extra $100. For this $100 investment, you will get an extra $70 in
the first year and $60 in the second year. Is this a good move?
Based on our discussion, the NPV of the switch, NPV(B—A) is:
NPV(B—A) = —100 +[70/(1 + R)] + [60/(1 + R)2]
We can calculate the return on this investment by setting the NPV equal to zero and solving
for the IRR:
NPV(B—A) = 0 = —100 +[70/(1 + R)] + [60/(1 + R)2]
If you go through this calculation, you will find the IRR is exactly 20 percent. What this tells
us is that at a 20 percent discount rate, we are indifferent between the two investments because the
NPV of the difference in their cash flows is zero. As a consequence, the two investments have the
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same value, so this 20 percent is the crossover rate. Check to see that the NPV at 20 percent is
$2,78 for both investments.
The Profitability Index
The profitability index (PI), or benefit-cost ratio, is defined as the present value of the future cash
flows divided by the initial investment. So, if a project cost $200 and the present value of its future
cash flows is $220, the profitability index value would be 220/200 = 1.1.
The profitability index would be bigger than 1 for a positive investment and less than 1 for a
negative NPV investment. PI is often proposed as a measure of performance for government or
other not-for-profit investment. Consider an investment that costs $5 and has a $10 present value
and an investment that costs $100 with a $150 present value. The first of these investments has an
NPV of $5 and a PI of 2. The second has an NPV of $50 and a PI of 1.5. If these are mutually
exclusive investments, then the second one is preferred even though it has a lower PI.
Modified Internal Rate of Return (MIRR)
Business executives often prefer to work with percentage rate of return, such as IRR, rather than
dollar amounts of NPV when analyzing investments. To overcome some of the IRR’s limitations a
Modified IRR, or MIRR, has been devised. The MIRR is defined as the discount rate that forces
PV costs = PV terminal value, where terminal value (TV) is the future value of the inflows
compounded at the project’s cost of capital. Thus
n
n
COFt


t
t  0 (1  k )
 CIF (1  k )
t 0
n t
t
(1  MIRR ) n
TV
PV costs =
(1+MIRR)n
MIRR assumes that cash flows are reinvested at the cost of capital rather than the project’s
own IRR, making it better indicator of a project’s true profitability.
Problems
1. A proposed overseas expansion has the following cash flows:
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Year
0
1
2
3
4
Cash Flow
-200
50
60
70
200
Calculate the payback, the discounted payback, and the NPV at a required return of 10 percent.
2. The director of capital budgeting for Giant Inc. has identified two mutually exclusive projects,
L and S, with the following expected net cash flows:
Expected Net Cash Flows (in millions)
Year
Project L
Project S
0
($100)
($100)
1
10
70
2
60
50
3
80
20
Both projects have a cost of capital of 10 percent.
(1) What is the payback period for Project S?
a. 1.6 years
b. 1.8 years
c. 2.1 years
d. 2.5 years
e. 2.8 years
c. $22.64
d. $18.78
e. $10.06
(2) What is Project L’s NPV?
a. $50.00
b. $34.25
(3) What is Project L’s IRR?
a. 18.1%
b. 19.7%
c. 21.4%
d. 23.6%
e. 24.2%
c. 16.9%
d. 17.1%
e. 17.4%
c. 16.9%
d. 17.1%
e. 17.4%
(4) What is the Problem L’s MIRR?
a. 15.3%
b. 16.5%
(5) What is Project S’s MIRR?
a. 15.3%
b. 16.5%
Answers to the Problems
1. In the following table, we have listed the required numbers for the proposed project.
Cash Flow
Year
Undiscounted
Accumulated Cash Flow
Discounted
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Undiscounted
Discounted
1
50
45.45
50
45.45
2
60
49.59
110
95.04
3
70
52.59
180
147.63
4
200
136.60
380
284.23
We see that payback occurs between Year 3 and year 4. The cash flows for the first three
years are $180 total, so, going into the fourth year, we are short by $20. The total cash flow in year
4 is $200, so the payback is 3 + (20/200) =3.10 years. Looking at the accumulated discounted cash
flows, we see that the discounted payback occurs between years 3 and Year 4. The sum of the
discounted cash flows is $284.23. So the NPV is $84.23.
2. Answer:
(1) a. After the first year, there is only $30 remaining to be paid, and $50 is received in Year 2.
Assuming an even cash flow throughout the years, the payback period is 1+30/50 = 1.6
years. d.
(2) d NPVL = —100 +10 /1.10 + 60 / (1.10)2 + 80 / (1.10)3 =18.79.
n
(3) a.
CFt
 (1  IRR )
t 0
t
=0 →
 100
10
60
80



0
0
1
2
(1  IRR )
(1  IRR )
(1  IRR )
(1  IRR ) 3
With a financial calculator, you can get IRR=18.1%
n
n
(4) b.
COFt
 (1  k )
t 0
100 
t

 CIF (1  k )
t 0
n t
t
(1  MIRR )
n
→
PV cos ts 
10(1.10) 2  60(1.10)1  80(1.10) 0
(1  MIRR ) 3
100 
TV
(1  M I R R
)n
158.10
(1  MIRR ) 3
MIRRL = 16.50%
(5) c. 100 
70(1.10) 2  50(1.10)1  20(1.10) 0
(1  MIRR ) 3
100 
159.70
(1  MIRR ) 3
MIRRL = 16.89%
Reference Book
Stephen A. Ross, Randolph W. Westerfield, Bradford D. Jordan: Fundamentals of Corporate
Finance, Fifth Edition, McGraw-Hill 2002.
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