ENGR 111 Fall 2011
HOMEWORK 3
ANSWER KEY
1.
We can apply the growing perpetuity formula to find the PV of stream
X
.
The perpetuity
formula values the stream as of one year before the first payment.
Therefore, the growing
perpetuity formula values the stream of cash flows as of year 2.
Next, discount the PV as of the
end of year 2 back two years to find the PV as of today, year 0. Doing so, we find:
PV(X) = [C
3
/ (R – g)] / (1 + R)
2
PV(X) = [$8,900 / (0.12 – 0.04)] / (1.12)
2
PV(X) = $89,684.31
We can apply the perpetuity formula to find the PV of stream
Y
.
The perpetuity formula
discounts the stream back to year 1, one period prior to the first cash flow.
Discount the PV as of
the end of year 1 back one year to find the PV as of today, year 0. Doing so, we find:
PV(Y) = [C
2
/ R] / (1 + R)
PV(Y) = [–$12,000 / 0.12] / (1.12)
PV(Y) = –$89,285.71
If we combine the cash flow streams to form Project C, we get:
Project X = [C
3
/ (R – G)] / (1 + R)
2
Project Y = [C
2
/ R] / (1 + R)
Project Z = Project
X
+ Project
Y
Project Z = [C
3
/ (R – g)] / (1 + R)
2
+ [C
2
/ R] / (1 +R)
0 = [$9,000 / (IRR – .04)] / (1 + IRR)
2
+ [–$12,000 / IRR] / (1 + IRR)
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
IRR = 12.09%
The correct decision rule for an investingtype project is to accept the project if the discount rate
is below the IRR.
Since there is one IRR, a decision can be made.
At a point in the future, the
cash flows from stream
X
will be greater than those from stream
Y
.
In year 11 the cash flow is
going to turn to positive from negative. In addition, in year 20, there will be a one time negative
cash flow which makes the changes in signs more than one.
When the sign of the cash flows change more than once over the life of the project, there may be
multiple internal rates of return. In such cases, there is no correct decision rule for accepting and
rejecting projects using the internal rate of return.
2.
The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR for
each project is:
0 = C
0
+ C
1
/ (1 + IRR) + C
2
/ (1 + IRR)
2
+ C
3
/ (1 + IRR)
3
0 = –$750 + $310 / (1 + IRR) + $430 / (1 + IRR)
2
+ $330 / (1 + IRR)
3
IRR = 19.83%
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0 = C
0
+ C
1
/ (1 + IRR) + C
2
/ (1 + IRR)
2
+ C
3
/ (1 + IRR)
3
0 = –$2,100 + $1,200/ (1 + IRR) + $760 / (1 + IRR)
2
+ $850 / (1 + IRR)
3
IRR = 17.36%
Based on the IRR rule, the first project should be chosen because it has the higher IRR.
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 Fall '11
 MelihaBuluTaciroglu
 Net Present Value, Internal rate of return, aftertax salvage value

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