April 3, 2019

CHAPTER 9 B-163

b. The equation for the IRR of the project is:

0 = –$45,000,000 + $78,000,000/(1+IRR) – $14,000,000/(1+IRR)2

From Descartes rule of signs, we know there are potentially two IRRs since the cash flows change

signs twice. From trial and error, the two IRRs are:

IRR = 53.00%, –79.67%

When there are multiple IRRs, the IRR decision rule is ambiguous. Both IRRs are correct, that is,

both interest rates make the NPV of the project equal to zero. If we are evaluating whether or not

to accept this project, we would not want to use the IRR to make our decision.

15. The profitability index is defined as the PV of the cash inflows divided by the PV of the cash outflows.

The equation for the profitability index at a required return of 10 percent is:

PI = [$7,300/1.1 + $6,900/1.12 + $5,700/1.13] / $14,000 = 1.187

The equation for the profitability index at a required return of 15 percent is:

PI = [$7,300/1.15 + $6,900/1.152 + $5,700/1.153] / $14,000 = 1.094

The equation for the profitability index at a required return of 22 percent is:

PI = [$7,300/1.22 + $6,900/1.222 + $5,700/1.223] / $14,000 = 0.983

We would accept the project if the required return were 10 percent or 15 percent since the PI is greater

than one. We would reject the project if the required return were 22 percent since the PI is less than one.

16. a. The profitability index is the PV of the future cash flows divided by the initial investment. The cash

flows for both projects are an annuity, so:

b. The NPV of each project is:

NPVI = –$53,000 + $27,000(PVIFA10%,3) = $14,145.00

B-164 SOLUTIONS

c. Using the profitability index to compare mutually exclusive projects can be ambiguous when the

17. a. The payback period for each project is:

A: 3 + ($180,000/$390,000) = 3.46 years

b. The discounted payback for each project is:

A: $20,000/1.15 + $50,000/1.152 + $50,000/1.153 = $88,074.30

$390,000/1.154 = $222,983.77

c. The NPV for each project is:

A: NPV = –$300,000 + $20,000/1.15 + $50,000/1.152 + $50,000/1.153 + $390,000/1.154

NPV = $11,058.07

B: NPV = –$40,000 + $19,000/1.15 + $12,000/1.152 + $18,000/1.153 + $10,500/1.154

d. The IRR for each project is:

A: $300,000 = $20,000/(1+IRR) + $50,000/(1+IRR)2 + $50,000/(1+IRR)3 + $390,000/(1+IRR)4

e. The profitability index for each project is:

A: PI = ($20,000/1.15 + $50,000/1.152 + $50,000/1.153 + $390,000/1.154) / $300,000 = 1.037

f. In this instance, the NPV criteria implies that you should accept project A, while profitability index,

payback period, discounted payback, and IRR imply that you should accept project B. The final

18. At a zero discount rate (and only at a zero discount rate), the cash flows can be added together across

time. So, the NPV of the project at a zero percent required return is:

NPV = –$684,680 + 263,279 + 294,060 + 227,604 + 174,356 = $274,619

If the required return is infinite, future cash flows have no value. Even if the cash flow in one year is $1

trillion, at an infinite rate of interest, the value of this cash flow today is zero. So, if the future cash

flows have no value today, the NPV of the project is simply the cash flow today, so at an infinite

interest rate:

B-166 SOLUTIONS

19. The MIRR for the project with all three approaches is:

Discounting approach:

In the discounting approach, we find the value of all cash outflows to time 0, while any cash inflows

remain at the time at which they occur. So, the discounting the cash outflows to time 0, we find:

Time 0 cash flow = –$16,000 – $5,100 / 1.105

Time 0 cash flow = –$19,166.70

So, the MIRR using the discounting approach is:

0 = –$19,166.70 + $6,100/(1+MIRR) + $7,800/(1+MIRR)2 + $8,400/(1+MIRR)3 + 6,500/(1+MIRR)4

Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:

MIRR = 18.18%

Reinvestment approach:

In the reinvestment approach, we find the future value of all cash except the initial cash flow at the end

of the project. So, reinvesting the cash flows to time 5, we find:

Time 5 cash flow = $6,100(1.104) + $7,800(1.103) + $8,400(1.102) + $6,500(1.10) – $5,100

Time 5 cash flow = $31,526.81

So, the MIRR using the discounting approach is:

0 = –$16,000 + $31,526.81/(1+MIRR)5

20. With different discounting and reinvestment rates, we need to make sure to use the appropriate interest

rate. The MIRR for the project with all three approaches is:

Discounting approach:

In the discounting approach, we find the value of all cash outflows to time 0 at the discount rate, while

any cash inflows remain at the time at which they occur. So, the discounting the cash outflows to time 0,

we find:

Time 0 cash flow = –$16,000 – $5,100 / 1.115

Time 0 cash flow = –$19,026.60

So, the MIRR using the discounting approach is:

0 = –$19,026.60 + $6,100/(1+MIRR) + $7,800/(1+MIRR)2 + $8,400/(1+MIRR)3 + 6,500/(1+MIRR)4

Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:

MIRR = 18.55%

Reinvestment approach:

In the reinvestment approach, we find the future value of all cash except the initial cash flow at the end

of the project using the reinvestment rate. So, the reinvesting the cash flows to time 5, we find:

Time 5 cash flow = $6,100(1.084) + $7,800(1.083) + $8,400(1.082) + $6,500(1.08) – $5,100

Time 5 cash flow = $29,842.50

21. Since the NPV index has the cost subtracted in the numerator, NPV index = PI – 1.

22. a. To have a payback equal to the project’s life, given C is a constant cash flow for N years:

C = I/N

b. To have a positive NPV, I < C (PVIFAR%, N). Thus, C > I / (PVIFAR%, N).

c. Benefits = C (PVIFAR%, N) = 2 × costs = 2I

C = 2I / (PVIFAR%, N)

Challenge

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:

24. The equation for the IRR of the project is:

0 = –$1,512 + $8,586/(1 + IRR) – $18,210/(1 + IRR)2 + $17,100/(1 + IRR)3 – $6,000/(1 + IRR)4

Using Descartes rule of signs, from looking at the cash flows we know there are four IRRs for this

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

perpetuity. If you remember back to the chapter on stock valuation, we presented a formula for

b. Here we want to know the minimum growth rate in cash flows necessary to accept the project. The

minimum growth rate is the growth rate at which we would have a zero NPV. The equation for a

zero NPV, using the equation for the PV of a growing perpetuity is:

26. The IRR of the project is:

$58,000 = $34,000/(1+IRR) + $45,000/(1+IRR)2

Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:

IRR = 22.14%

At an interest rate of 12 percent, the NPV is:

NPV = $58,000 – $34,000/1.12 – $45,000/1.122

NPV = –$8,230.87

At an interest rate of zero percent, we can add cash flows, so the NPV is:

NPV = $58,000 – $34,000 – $45,000

B-170 SOLUTIONS

27. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR of the project

is:

0 = $20,000 – $26,000 / (1 + IRR) + $13,000 / (1 + IRR)2

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 the IRR

equation, what we are really doing is solving for the roots of the equation. Going back to high school

algebra, in this problem we are solving a quadratic equation. In case you don’t remember, the quadratic

equation is:

x = a

acbb

2

4

2

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 = $205,000(1.04) = $213,200

Year 3 cash flow = $265,000(1.04) = $275,600

Year 4 cash flow = $346,000(1.04) = $359,840

Year 5 cash flow = $220,000(1.04) = $228,800

So, the NPV of the project is:

NPV = –$450,000 + $213,200/1.112 + $275,600/1.113 + $359,840/1.114 + $228,800/1.115

Calculator Solutions

7.

CFo –$34,000

C01 $16,000

F01 1

C02 $18,000

8.

CFo –$34,000 CFo –$34,000

C01 $16,000 C01 $16,000

F01 1 F01 1

C02 $18,000 C02 $18,000

F02 1 F02 1

9.

CFo –$138,000 CFo –$138,000 CFo –$138,000

C01 $28,500 C01 $28,500 C01 $28,500

F01 9 F01 9 F01 9