23. Given the seven-year payback, the worst case is that the payback occurs at the end of the seventh
year. Thus, the worst-case:
The best case has infinite cash flows beyond the payback point. Thus, the best-case NPV is infinite.
24. The equation for the IRR of the project is:
Using Descartes rule of signs, from looking at the cash flows we know there are four IRRs for this
We would accept the project when the NPV is greater than zero. See for yourself that the NPV is
25. a. Here the cash inflows of the project go on forever, which is a perpetuity. Unlike ordinary
perpetuity cash flows, the cash flows here grow at a constant rate forever, which is a growing
PV of cash inflows = C1/(R – g)
NPV is the PV of the inflows minus the PV of the outflows, so the NPV is:
The NPV is positive, so we would accept the project.
b. Here we want to know the minimum growth rate in cash flows necessary to accept the project.
Solving for g, we get:
26. The IRR of the project is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
At an interest rate of 12 percent, the NPV is:
At an interest rate of zero percent, we can add cash flows, so the NPV is:
And at an interest rate of 24 percent, the NPV is:
The cash flows for the project are unconventional. Since the initial cash flow is positive and the
27. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR of the
project is:
Even though it appears there are two IRRs, a spreadsheet, financial calculator, or trial and error will
not give an answer. The reason is that there is no real IRR for this set of cash flows. If you examine
x =
−b±
√
b2−4ac
2a
In this case, the equation is:
x =
−(−11,000 )±
√
(−11,000 )2−4(7, 000 )(25 ,000 )
2(7, 000 )
The square root term works out to be:
The square root of a negative number is a complex number, so there is no real number solution,
meaning the project has no real IRR.
28. First, we need to find the future value of the cash flows for the one year in which they are blocked by
the government. So, reinvesting each cash inflow for one year, we find:
Year 2 cash flow = $435,000(1.04) = $452,400
So, the NPV of the project is:
And the IRR of the project is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
While this may look like a MIRR calculation, it is not a MIRR, rather it is a standard IRR
Calculator Solutions
7.
CFo–$34,000
C01 $15,000
F01 1
F03 1
IRR CPT
8.
CFo–$34,000 CFo–$34,000
9.
CFo–$63,000 CFo–$63,000 CFo–$63,000
10.
CFo–$15,400
C01 $7,300
F01 1
11.
CFo–$15,400 CFo–$15,400
C01 $7,300 C01 $7,300
F01 1F01 1
CFo–$15,400 CFo–$15,400
C01 $7,300 C01 $7,300
F01 1F01 1
12. Project A
CFo–$37,500 CFo–$37,500
C01 $17,300 C01 $17,300
F01 1F01 1
$6,330.67
Project B
CFo–$37,500 CFo–$37,500
C01 $5,700 C01 $5,700
F01 1F01 1
$8,138.59
Crossover rate
CFo$0
C01 $11,600
F01 1
13. Project X
CFo–$23,000 CFo–$23,000 CFo–$23,000
C01 $10,490 C01 $10,490 C01 $10,490
F01 1F01 1F01 1
Project Y
CFo–$23,000 CFo–$23,000 CFo–$23,000
C01 $12,000 C01 $12,000 C01 $12,000
F01 1F01 1F01 1
Crossover rate
CFo$0
C01 –$1,510
F01 1
14.
CFo–$48,000,000 CFo–$48,000,000
Financial calculators will only give you one IRR, even if there are multiple IRRs. Using trial and
error, or a root solving calculator, the other IRR is –78.59%.
15.
CFo$0 CFo$0 CFo$0
C01 $9,700 C01 $9,700 C01 $9,700
F01 1F01 1F01 1
16. Project I
CFo$0 CFo–$63,000
Project II
CFo$0 CFo–$15,500
17.
CF(A) c. d. e.
CFo–$364,000 CFo–$364,000 CFo$0
C01 $46,000 C01 $46,000 C01 $46,000
F01 1F01 1F01 1
CF(B) c. d. e.
CFo–$52,000 CFo–$52,000 CFo$0
C01 $25,000 C01 $25,000 C01 $25,000
F01 1F01 1F01 1
18.
CFo–$527,630 CFo–$527,630
C01 $212,200 C01 $212,200
F01 1F01 1
26.
CFo$59,000 CFo$59,000 CFo$59,000
C01 –$34,000 C01 –$34,000 C01 –$34,000
F01 1F01 1F01 1
CFo$59,000
C01 –$34,000
F02 1
IRR CPT