1. To calculate the payback period, we need to find the time that the project has recovered its initial investment. After two years, the project has created:
$1,200 + 2,500 = $3,700
in cash flows. The project still needs to create another:
$4,800 – 3,700 = $1,100
in cash flows. During the third year, the cash flows from the project will be $3,400. So, the payback period will be 2 years, plus what we still need to make divided by what we will make during the third year. The payback period is:
Payback = 2 + ($1,100 / $3,400) = 2.32 years
2. To calculate the payback period, we need to find the time that the project has recovered its initial investment. The cash flows in this problem are an annuity, so the calculation is simpler. If the initial cost is $3,000, the payback period is:
Payback = 3 + ($480 / $840) = 3.57 years
There is a shortcut to calculate the future cash flows are an annuity. Just divide the initial cost by the annual cash flow. For the $3,000 cost, the payback period is:
Payback = $3,000 / $840 = 3.57 years
For an initial cost of $5,000, the payback period is:
Payback = 5 + ($800 / $840) = 5.95 years
The payback period for an initial cost of $7,000 is a little trickier. Notice that the total cash inflows after eight years will be:
Total cash inflows = 8($840) = $6,720
If the initial cost is $7,000, the project never pays back. Notice that if you use the shortcut for annuity cash flows, you get:
Payback = $7,000 / $840 = 8.33 years.
This answer does not make sense since the cash flows stop after eight years, so again, we must conclude the payback period is never.
3. Project A has cash flows of:
Cash flows = $30,000 + 18,000 =$48,000
during this first two years. The cash flows are still short by $2,000 of recapturing the initial investment, so the payback for Project A is:
Payback = 2 + ($2,000 / $10,000) = 2.20 years
Project B has cash flows of:
Cash flows = $9,000 + 25,000 + 35,000 =$69,000
during this first two years. The cash flows are still short by $1,000 of recapturing the initial investment, so the payback for Project B is:
B: Payback = 3 + ($1,000 / $425,000) = 3.002 years
Using the payback criterion and a cutoff of 3 years, accept project A and reject project B.
4. When we use discounted payback, we need to find the value of all cash flows today. The value today of the project cash flows for the first four years is:
Value today of Year 1 cash flow = $7,000/1.14 = $6,140.35
Value today of Year 2 cash flow = $7,500/1.142 = $5,771.01
Value today of Year 3 cash flow = $8,000/1.143 = $5,399.77
Value today of Year 4 cash flow = $8,500/1.144 = $5,032.68
To find the discounted payback, we use these values to find the payback period. The discounted first year cash flow is $6,140.35, so the discounted payback for an $8,000 initial cost is:
Discounted payback = 1 + ($8,000 – 6,140.35)/$5,771.01 = 1.32 years
For an initial cost of $13,000, the discounted payback is:
Discounted payback = 2 + ($13,000 – 6,140.35 – 5,771.01)/$5,399.77 = 2.20 years
Notice the calculation of discounted payback. We know the payback period is between two and three years, so we subtract the discounted values of the Year 1 and Year 2 cash flows from the initial cost. This is the numerator, which is the discounted amount we still need to make to recover our initial investment. We divide this amount by the discounted amount we will earn in Year 3 to get the fractional portion of the discounted payback.
If the initial cost is $18,000, the discounted payback is:
Discounted payback = 3 + ($18,000 – 6,140.35 – 5,771.01 – 5,399.77) / $5,032.68 = 3.14 years
5. R = 0%: 4 + ($1,600 / $2,100) = 4.76 years
discounted payback = regular payback = 4.76 years
R = 5%: $2,100/1.05 + $2,100/1.052 + $2,100/1.053 + $2,100/1.054 + $2,100/1.055 = $9,091.90
$2,100/1.056 = $1,567.05
discounted payback = 5 + ($10,000 – 9,091.90) / $1,567.05 = 5.58 years
R = 15%: $2,100/1.15 + $2,100/1.152 + $2,100/1.153 + $2,100/1.154 + $2,100/1.155 + $2,100/1.156 = $7,947.41; The project never pays back.
6. Our definition of AAR is the average net income divided by the average book value. The average net income for this project is:
Average net income = ($1,416,000 + 1,868,000 + 1,562,000 + 985,000) / 4 = $1,457,750
And the average book value is:
Average book value = ($15M + 0) / 2 = $7.5M
So, the AAR for this project is:
AAR = Average net income / Average book value = $1,457,750 / $7,500,000 = 19.44%
7. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines the IRR for this project is:
0 = – $30,000 + $20,000/(1+IRR) + $14,000/(1+IRR)2 + $11,000/(1+IRR)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
IRR = 26.48%
Since the IRR is greater than the required return we would accept the project.
8. The NPV of a project is the PV of the outflows minus by the PV of the inflows. The equation for the NPV of this project at an 11 percent required return is:
NPV = – $30,000 + $20,000/1.11 + $14,000/1.112 + $11,000/1.113 = $7,423.84
At an 11 percent required return, the NPV is positive, so we would accept the project.
The equation for the NPV of the project at a 30 percent required return is:
NPV = – $30,000 + $20,000/1.30 + $14,000/1.302 + $11,000/1.303 = – $1,324.53
At a 30 percent required return, the NPV is negative, so we would reject the project.
9. The NPV of a project is the PV of the outflows minus by the PV of the inflows. Since the cash inflows are an annuity, the equation for the NPV of this project at an 8 percent required return is:
NPV = – $70,000 + $14,000(PVIFA8%, 9) = $17,456.43
At an 8 percent required return, the NPV is positive, so we would accept the project.
The equation for the NPV of the project at a 16 percent required return is:
NPV = – $70,000 + $14,000(PVIFA16%, 9) = –$5,508.39
At a 16 percent required return, the NPV is negative, so we would reject the project.
We would be indifferent to the project if the required return was equal to the IRR of the project, since at that required return the NPV is zero. The IRR of the project is:
0 = – $40,000 + $14,000(PVIFAIRR, 9)
IRR = 13.70%
10. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines the IRR for this project is:
0 = – $8,000 + $3,200/(1+IRR) + $4,000/(1+IRR)2 + $6,100/(1+IRR)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
IRR = 26.83%
11. The NPV of a project is the PV of the outflows minus by the PV of the inflows. 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 = – $8,000 + 3,200 + 4,000 + 6,100 = $5,300
The NPV at a 10 percent required return is:
NPV = – $8,000 + $3,200/1.1 + $4,000/1.12 + $6,100/1.13 = $2,797.90
The NPV at a 20 percent required return is:
NPV = – $8,000 + $3,200/1.2 + $4,000/1.22 + $6,100/1.23 = $974.54
And the NPV at a 30 percent required return is:
NPV = – $8,000 + $3,200/1.3 + $4,000/1.32 + $6,100/1.33 = – $395.08
Notice that as the required return increases, the NPV of the project decreases. This will always be true for projects with conventional cash flows. Conventional cash flows are negative at the beginning of the project and positive throughout the rest of the project.
12. a. The IRR is the interest rate that makes the NPV of the project equal to zero. The equation for the IRR of Project A is:
0 = –$34,000 + $16,500/(1+IRR) + $14,000/(1+IRR)2 + $10,000/(1+IRR)3 + $6,000/(1+IRR)4
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
IRR = 16.60%
The equation for the IRR of Project B is:
0 = –$34,000 + $5,000/(1+IRR) + $10,000/(1+IRR)2 + $18,000/(1+IRR)3 + $19,000/(1+IRR)4
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
IRR = 15.72%
Examining the IRRs of the projects, we see that the IRRA is greater than the IRRB, so IRR decision rule implies accepting project A. This may not be a correct decision; however, because the IRR criterion has a ranking problem for mutually exclusive projects. To see if the IRR decision rule is correct or not, we need to evaluate the project NPVs.
b. The NPV of Project A is:
NPVA = –$34,000 + $16,500/1.11+ $14,000/1.112 + $10,000/1.113 + $6,000/1.114
NPVA = $3,491.88
And the NPV of Project B is:
NPVB = –$34,000 + $5,000/1.11 + $10,000/1.112 + $18,000/1.113 + $19,000/1.114
NPVB = $4,298.06
The NPVB is greater than the NPVA, so we should accept Project B.
c. To find the crossover rate, we subtract the cash flows from one project from the cash flows of the other project. Here, we will subtract the cash flows for Project B from the cash flows of Project A. Once we find these differential cash flows, we find the IRR. The equation for the crossover rate is:
Crossover rate: 0 = $11,500/(1+R) + $4,000/(1+R)2 – $8,000/(1+R)3 – $13,000/(1+R)4
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
R = 13.75%
At discount rates above 13.75% choose project A; for discount rates below 13.75% choose project B; indifferent between A and B at a discount rate of 13.75%.
13. The IRR is the interest rate that makes the NPV of the project equal to zero. The equation to calculate the IRR of Project X is:
0 = –$5,000 + $2,700/(1+IRR) + $1,700/(1+IRR)2 + $2,300/(1+IRR)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
IRR = 16.82%
For Project Y, the equation to find the IRR is:
0 = –$5,000 + $2,300/(1+IRR) + $1,800/(1+IRR)2 + $2,700/(1+IRR)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
IRR = 16.60%
To find the crossover rate, we subtract the cash flows from one project from the cash flows of the other project, and find the IRR of the differential cash flows. We will subtract the cash flows from Project Y from the cash flows from Project X. It is irrelevant which cash flows we subtract from the other. Subtracting the cash flows, the equation to calculate the IRR for these differential cash flows is:
Crossover rate: 0 = $400/(1+R) – $100/(1+R)2 – $400/(1+R)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
R = 13.28%
The table below shows the NPV of each project for different required returns. Notice that Project Y always has a higher NPV for discount rates below 13.28 percent, and always has a lower NPV for discount rates above 13.28 percent.
| R | $NPVX | $NPVY | |
| 0% | 1,700.00 | 1,800.00 | |
| 5% | 1,100.21 | 1,155.49 | |
| 10% | 587.53 | 607.06 | |
| 15% | 145.56 | 136.35 | |
| 20% | (238.43) | (270.83) | |
| 25% | (574.40) | (625.60) |
14. a. The equation for the NPV of the project is:
NPV = – $28M + $53M/1.1 – $8M/1.12 = $13,570,247.93
The NPV is greater than 0, so we would accept the project.
b. The equation for the IRR of the project is:
0 = –$28M + $53M/(1+IRR) – $8M/(1+IRR)2
From Descartes rule of signs, we know there are two IRRs since the cash flows change signs twice. From trial and error, the two IRRs are:
IRR = 72.75%, –83.46%
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 = [$3,200/1.1 + $3,900/1.12 + $2,600/1.13] / $7,000 = 1.155
The equation for the profitability index at a required return of 15 percent is:
PI = [$3,200/1.15 + $3,900/1.152 + $2,600/1.153] / $7,000 = 1.063
The equation for the profitability index at a required return of 22 percent is:
PI = [$3,200/1.22 + $3,900/1.222 + $2,600/1.223] / $7,000 = 0.954
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:
PII = $15,000(PVIFA10%,3 ) / $30,000 = 1.243
PIII = $2,800(PVIFA10%,3) / $5,000 = 1.393
The profitability index decision rule implies that we accept project II, since PIII is greater than the PII.
b. The NPV of each project is:
NPVI = – $30,000 + $15,000(PVIFA10%,3) = $7,302.78
NPVII = – $5,000 + $2,800(PVIFA10%,3) = $1,963.19
The NPV decision rule implies accepting Project I, since the NPVI is greater than the NPVII.
c. Using the profitability index to compare mutually exclusive projects can be ambiguous when the magnitude of the cash flows for the two projects are of different scale. In this problem, project I is roughly 3 times as large as project II and produces a larger NPV, yet the profit-ability index criterion implies that project II is more acceptable.
17. a. The payback period for each project is:
A: 3 + ($135K/$370K) = 3.36 years
B: 2 + ($1K/$11K) = 2.09 years
The payback criterion implies accepting project B, because it pays back sooner than project A.
b. The discounted payback for each project is:
A: $15K/1.15 + $30K/1.152 + $30K/1.153 = $55,453.28
$370K/1.154 = $211,548.70
Discounted payback = 3 + ($210,000 – 55,453.28)/$211,548.70 = 3.73 years
B: $11K/1.15 + $9K/1.152 = $16,370.51
$11K/1.153 = $7,232.68
Discounted payback = 2 + ($21,000 – 16,370.51)/$7,232.68 = 2.64 years
The discounted payback criterion implies accepting project B because it pays back sooner than A
c. The NPV for each project is:
A: NPV = – $210K + $15K/1.15 + $30K/1.152 + $30K/1.153 + $370K/1.154 = $57,001.98
B: NPV = – $21K + $11K/1.15 + $9K/1.152 + $11K/1.153 + $9K/1.154 = $7,748.97
NPV criterion implies we accept project A because project A has a higher NPV than project B.
d. The IRR for each project is:
A: $210K = $15K/(1+IRR) + $30K/(1+IRR)2 + $30K/(1+IRR)3 + $370K/(1+IRR)4
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
IRR = 22.97%
B: $21K = $11K/(1+IRR) + $9K/(1+IRR)2 + $11K/(1+IRR)3 + $9K/(1+IRR)4
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
IRR = 32.73%
IRR decision rule implies we accept project B because IRR for B is greater than IRR for A.
e. The profitability index for each project is:
A: PI = ($15K/1.15 + $30K/1.152 + $30K/1.153 + $370K/1.154) / $210K = 1.271
B: PI = ($11K/1.15 + $9K/1.152 + $11K/1.153 + $9K/1.154) / $21K = 1.369
Profitability index criterion implies accept project A because its PI is greater than project B’s.
f. In this instance, the NPV and PI criterion imply that you should accept project A, while payback period, discounted payback and IRR imply that you should accept project B. The final decision should be based on the NPV since it does not have the ranking problem associated with the other capital budgeting techniques. Therefore, you should accept project A.
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 = – $568,240 + 289,348 + 196,374 + 114,865 + 93,169 = $125,516
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:
NPV = – $568,240
The interest rate that makes the NPV of a project equal to zero is the IRR. The equation for the IRR of this project is:
0 = –$568,240 + $289,348/(1+IRR) + $196,374/(1+IRR)2 + $114,865/(1+IRR)3 + $93,169/(1+IRR)4
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
IRR = 10.71%
Intermediate
19. Since the NPV index has the cost subtracted in the numerator, NPV index = PI – 1.
20. 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
21. Given the seven year payback, the worst case is that the payback occurs at the end of the seventh year. Thus, the worst-case:
NPV = –$483,000 + $483,000/1.127 = –$264,515.33
The best case has infinite cash flows beyond the payback point. Thus, the best-case NPV is infinite.
22. 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%.
23. 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%
