What are the IRRs for the project?
What discount rate results in the maximum NPV for this
project? How can you determine that this is the maximum NPV?
(the highlighted part is
NOT required)
Ans :
The equation for the IRR of the project is:
0 = –$75,000 + $155,000 / (1 + IRR) – $65,000 / (1 + IRR)
2
From Descartes’ Rule of Signs, we know there are either zero IRRs or two IRRs since the cash
flows change signs twice. We can rewrite this equation as:
0 = –$75,000 + $155,000X – $65,000X
2
where X = 1 / (1 + IRR)
(the highlighted part of the solution is NOT required)
This is a quadratic equation. We can solve for the roots of this equation with the quadratic
formula:
X =
−
b
±
√
b
2
−
4
ac
2
a
Remember that the quadratic formula is written as:
0 =
a
X
2
+
b
X +
c
In this case, the equation is:
0 = –$65,000X
2
+ $155,000X – $75,000
X =
−
155
,
000
±
√
(
155,000
)
2
−
4
(−
75
,
000
)(−
65
,
000
)
2
(−
65
,
000
)
X =
−
155
,
000
±
√
4,525,000,000
2
(−
65
,
000
)
X =
−
155
,
000
±
67
,
268.12
−
130
,
000
Solving the quadratic equation, we find two Xs:
X = .6749, 1.7098
Since:
X = 1 / (1 + IRR)
1.7098 = 1 / (1 + IRR)
IRR = –.4151, or – 41.51%
5

Capital budgeting
And:
X = 1 / (1 + IRR)
.6749 = 1 / (1 + IRR)
IRR = .4818, or 48.18%
To find the maximum (or minimum) of a function, we find the derivative and set it equal to zero.
The derivative of this IRR function is:
0 = –$155,000(1 + IRR)
–2
+ $130,000(1 + IRR)
–3
–$155,000(1 + IRR)
–2
= $130,000(1 + IRR)
–3
–$155,000(1 + IRR)
3
= $130,000(1 + IRR)
2
–$155,000(1 + IRR) = $130,000
IRR = $130,000 / $155,000 – 1
IRR = – .1613, or –16.13%
To determine if this is a maximum or minimum, we can find the second derivative of the IRR
function. If the second derivative is positive, we have found a minimum and if the second derivative is
negative we have found a maximum. Using the reduced equation above, that is:
–$155,000(1 + IRR) = $130,000
The second derivative is –$262,722.18, therefore we have a maximum.
6

Capital budgeting
Chapter 6:
2. Incremental Cash Flows Which of the following should be treated as an incremental
cash flow when computing the NPV of an investment?
a. A reduction in the sales of a company’s other products caused by the investment.
Ans:
Yes, the reduction in the sales of the company’s other products, referred to as erosion, should be
treated as an incremental cash flow. These lost sales are included because they are a cost (a revenue
reduction) that the firm must bear if it chooses to produce the new product.
b. An expenditure on plant and equipment that has not yet been made and will be made
only if the project is accepted.
Ans
:
Yes, expenditures on plant and equipment should be treated as incremental cash flows.
These are costs of the new product line. However, if these expenditures have already
occurred (and cannot be recaptured through a sale of the plant and equipment), they are
sunk costs and are
not included as incremental cash flows.
C.Costs of research and development undertaken in connection with the product during
the past three years
No, the research and development costs
should not be treated as incremental cash flows.
The costs of research and development undertaken on the product during the past three
years are sunk costs and should not be included in the evaluation of the project. Decisions
made and costs incurred in the past cannot be changed. They should not affect the decision
to accept or reject the project.

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