NET PRESENT VALUE AND OTHER INVESTMENT CRITERIA
25. The equation for the IRR of the project is:
0 = –€504 + €2,862/(1 + IRR) – €6,070/(1 + IRR)2 + €5,700/(1 + IRR)3 – €2,000/(1 + IRR)4
Using Descartes rule of signs, from looking at the cash flows we know there are four IRRs for this project. Even with most computer spreadsheets, we have to do some trial and error. From trial and error, IRRs of 25%, 33.33%, 42.86%, and 66.67% are found.
We would accept the project when the NPV is greater than zero. See for yourself if that NPV is greater than zero for required returns between 25% and 33.33% or between 42.86% and 66.67%.
26. 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 valuing a stock with constant growth in dividends. This formula is actually the formula for a growing perpetuity, so we can use it here. The PV of the future cash flows from the project is:
PV of cash inflows = C1/(R – g)
PV of cash inflows = $50,000/(.13 – .06) = $714,285.71
NPV is the PV of the outflows minus by the PV of the inflows, so the NPV is:
NPV of the project = –$780,000 + 714,285.71 = –$65,714.29 The NPV is negative, so we would reject the project.
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:
0 = – $780,000 + $50,000/(.13 – g) Solving for g, we get:
g = 6.59%
27. a. The project involves three cash flows: the initial investment, the annual cash inflows, and the abandonment costs. The mine will generate cash inflows over its 11-year economic life. To express the PV of the annual cash inflows, apply the growing annuity formula, discounted at the IRR and growing at eight percent.
PV(Cash Inflows) = C {[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)]t}
PV(Cash Inflows) = R100,000{[1/(IRR – .08)] – [1/(IRR – .08)] × [(1 + .08)/(1 + IRR)]11} At the end of 11 years, the Great Karroo Mining Corporate will abandon the mine, incurring a R75,000 charge. Discounting the abandonment costs back 11 years at the IRR to express its present value, we get:
PV(Abandonment) = C11 / (1 + IRR)11 PV(Abandonment) = –R75,000 / (1+IRR)11
So, the IRR equation for this project is:
0 = –R600,000 + R100,000{[1/(IRR – .08)] – [1/(IRR – .08)] × [(1 + .08)/(1 + IRR)]11} –R75,000 / (1+IRR)11
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
IRR = 18.40%
b. Yes. Since the mine’s IRR exceeds the required return of 10 percent, the mine should be opened. The correct decision rule for an investment-type project is to accept the project if the discount rate is above the IRR. Although it appears there is a sign change at the end of the project because of the abandonment costs, the last cash flow is actually positive because the operating cash in the last year.
28. a. We can apply the growing perpetuity formula to find the PV of stream A. 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(A) = [C3 / (R – g)] / (1 + R)2
PV(A) = [€5,000 / (.12 – 0.04)] / (1.12)2 PV(A) = €49,824.62
We can apply the perpetuity formula to find the PV of stream B. 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:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
IRR = 14.65%
c. The correct decision rule for an investing-type 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 A will be greater than those from stream B. Therefore, although there are many cash flows, there will be only one change in sign. 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.
29. To answer this question, we need to examine the incremental cash flows. To make the projects equally attractive, Project Billion must have a larger initial investment. We know this because the subsequent cash flows from Project Billion are larger than the subsequent cash flows from Project Million. So, subtracting the Project Million cash flows from the Project Billion cash flows, we find the incremental cash flows are:
Now we can find the present value of the subsequent incremental cash flows at the discount rate, 12 percent. The present value of the subsequent incremental cash flows is:
PV = $300 / 1.12 + $300 / 1.122 + $500 / 1.123 PV = $862.91
To make the projects equally as attractive, we would need to make sure the incremental cash flows have a positive NPV, which implies a profitability index greater than one. Setting the profitability index of the incremental cash flows equal to one, we find:
PI = 1 = $862.91 / (–I0 + 1,500)
Solving for I0, the initial investment must be between $637.09 and $1,500
30. 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
The square root term works out to be:
676,000,000 – 1,040,000,000 = –364,000,000
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.
Calculator Solutions
1. b. Project A
CFo –€7,500 CFo –€5,000
C01 €4,000 C01 €2,500
F01 1 F01 1
C02 €3,500 C02 €1,200
F02 1 F02 1
C03 €1,500 C03 €3,000
F03 1 F03 1
I = 15% I = 15%
NPV CPT NPV CPT
–€388.96 €53.83
7.
CFo –¥8,000,000 C01 ¥4,000,000 F01 1
C02 ¥3,000,000 F02 1
C03 ¥2,000,000 F03 1
IRR CPT 6.93%
8. Project A Project B
CFo –£2,000 CFo –£1,500
C01 £1,000 C01 £500
F01 1 F01 1
C02 £1,500 C02 £1,000
F02 1 F02 1
C03 £2,000 C03 £1,500
F03 1 F03 1
IRR CPT IRR CPT
47.15% 36.19%
9.
13. a. Deepwater fishing Submarine ride
CFo –$600,000 CFo –$1,800,000
b.
c. Deepwater fishing Submarine ride
CFo –$600,000 CFo –$1,800,000
15.
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 –83.46%.
16. b. Board game CD-ROM
17. a. CDMA G4 Wi-Fi
38,880,540.95 89,556,724.27 126,371,149.51
PICDMA = 38,880,540.95 / 10,000,000 = 3.89
19. a. Project A Project B Project C
c. Akita Fukui
CFo –¥1,000,000 CFo –¥500,000
C01 ¥600,000 C01 ¥300,000
F01 1 F01 1
C02 ¥400,000 C02 ¥500,000
F02 1 F02 1
C03 ¥1,000,000 C03 ¥100,0000
F03 1 F03 1
IRR CPT IRR CPT
39.79% 40.99%
d.
CFo –¥500,000 C01 ¥300,000 F01 1
C02 –¥100,000 F02 1
C03 ¥900,000 F03 1
IRR CPT 38.90%
21. a. NP-30 NX-20
CFo –€100,000 CFo –€30,000
C01 €40,000 C01 €20,000
F01 5 F01 1
C02 C02 €23,000
F02 F02 1
C03 C03 €26,450
F03 F03 1
C04 C04 €30,418
F04 F04 1
C05 C05 €34,890
F05 F05 1
I = 15% I = 15%
NPV CPT NPV CPT
€34,086.20 €56,956.75
b. NP-30 NX-20
22. a. Project A Project B
d. Project A Project B
e. Project A Project B Project C
CFo –£150,000 CFo –£200,000 CFo –£100,000
C01 £50,000 C01 £200,000 C01 £100,000
F01 1 F01 1 F01 2
C02 £100,000 C02 £111,000 C02
F02 1 F02 1 F02
I = 10% I = 20% I = 20%
NPV CPT NPV CPT NPV CPT
–£21,900.83 £43,750.00 £52,777.78
30.
CFo ¥20,000 C01 –¥26,000 F01 1
C02 ¥13,000 F02 1 IRR CPT ERROR 7