1. In this context, an opportunity cost refers to the value of an asset or other input that will be used in a project. The relevant cost is what the asset or input is actually worth today, not, for example, what it cost to acquire.
2. For tax purposes, a firm would choose MACRS because it provides for larger depreciation deductions earlier. These larger deductions reduce taxes, but have no other cash consequences. Notice that the choice between MACRS and straight-line is purely a time value issue; the total depreciation is the same, only the timing differs.
3. It’s probably only a mild over-simplification. Current liabilities will all be paid, presumably. The cash portion of current assets will be retrieved. Some receivables won’t be collected, and some inventory will not be sold, of course. Counterbalancing these losses is the fact that inventory sold above cost (and not replaced at the end of the project’s life) acts to increase working capital. These effects tend to offset one another.
4. Management’s discretion to set the firm’s capital structure is applicable at the firm level. Since any one particular project could be financed entirely with equity, another project could be financed with debt, and the firm’s overall capital structure remains unchanged, financing costs are not relevant in the analysis of a project’s incremental cash flows according to the stand-alone principle.
5. The EAC approach is appropriate when comparing mutually exclusive projects with different lives that will be replaced when they wear out. This type of analysis is necessary so that the projects have a common life span over which they can be compared; in effect, each project is assumed to exist over an infinite horizon of N-year repeating projects. Assuming that this type of analysis is valid implies that the project cash flows remain the same forever, thus ignoring the possible effects of, among other things: (1) inflation, (2) changing economic conditions, (3) the increasing unreliability of cash flow estimates that occur far into the future, and (4) the possible effects of future technology improvement that could alter the project cash flows.
6. Depreciation is a non-cash expense, but it is tax-deductible on the income statement. Thus depreciation causes taxes paid, an actual cash outflow, to be reduced by an amount equal to the depreciation tax shield tcD. A reduction in taxes that would otherwise be paid is the same thing as a cash inflow, so the effects of the depreciation tax shield must be added in to get the total incremental aftertax cash flows.
7. There are two particularly important considerations. The first is erosion. Will the essentialized book simply displace copies of the existing book that would have otherwise been sold? This is of special concern given the lower price. The second consideration is competition. Will other publishers step in and produce such a product? If so, then any erosion is much less relevant. A particular concern to book publishers (and producers of a variety of other product types) is that the publisher only makes money from the sale of new books. Thus, it is important to examine whether the new book would displace sales of used books (good from the publisher’s perspective) or new books (not good). The concern arises any time there is an active market for used product.
8. Definitely. The damage to Porsche’s reputation is definitely a factor the company needed to consider. If the reputation was damaged, the company would have lost sales of its existing car lines.
9. One company may be able to produce at lower incremental cost or market better. Also, of course, one of the two may have made a mistake!
10. Porsche would recognize that the outsized profits would dwindle as more product comes to market and competition becomes more intense.
Solutions to Questions and Problems
NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem.
Basic
1. The $5 million acquisition cost of the land six years ago is a sunk cost. The $5.4 million current aftertax value of the land is an opportunity cost if the land is used rather than sold off. The $10.4 million cash outlay and $650,000 grading expenses are the initial fixed asset investments needed to get the project going. Therefore, the proper year zero cash flow to use in evaluating this project is
$5,400,000 + 10,400,000 + 650,000 = $16,450,000
2. Sales due solely to the new product line are:
21,000($12,000) = $252,000,000
Increased sales of the motor home line occur because of the new product line introduction; thus:
5,000($45,000) = $225,000,000
in new sales is relevant. Erosion of luxury motor coach sales is also due to the new mid-size campers; thus:
1,300($85,000) = $110,500,000 loss in sales
is relevant. The net sales figure to use in evaluating the new line is thus:
$252,000,000 + 225,000,000 – 110,500,000 = $366,500,000
3. We need to construct a basic income statement. The income statement is:
Sales $ 650,000
Variable costs 390,000
Fixed costs 158,000
Depreciation 75,000
EBT $ 27,000
[email protected]% 9,450
Net income $ 17,550
4. To find the OCF, we need to complete the income statement as follows:
Sales $ 912,400
Costs 593,600
Depreciation 135,000
EBT $ 183,800
Ta[email protected]% 62,492
Net income $ 121,308
The OCF for the company is:
OCF = EBIT + Depreciation – Taxes
OCF = $183,800 + 135,000 – 62,492
OCF = $256,308
The depreciation tax shield is the depreciation times the tax rate, so:
Depreciation tax shield = tcDepreciation
Depreciation tax shield = .34($135,000)
Depreciation tax shield = $45,900
The depreciation tax shield shows us the increase in OCF by being able to expense depreciation.
5. To calculate the OCF, we first need to calculate net income. The income statement is:
Sales $ 85,000
Variable costs 43,000
Depreciation 3,000
EBT $ 39,000
[email protected]% 13,650
Net income $ 25,350
Using the most common financial calculation for OCF, we get:
OCF = EBIT + Depreciation – Taxes = $39,000 + 3,000 – 13,650
OCF = $28,350
The top-down approach to calculating OCF yields:
OCF = Sales – Costs – Taxes = $85,000 – 43,000 – 13,650
OCF = $28,350
The tax-shield approach is:
OCF = (Sales – Costs)(1 – tC) + tCDepreciation
OCF = ($85,000 – 43,000)(1 – .35) + .35(3,000)
OCF = $28,350
And the bottom-up approach is:
OCF = Net income + Depreciation = $25,350 + 3,000
OCF = $28,350
All four methods of calculating OCF should always give the same answer.
6. The MACRS depreciation schedule is shown in Table 10.7. The ending book value for any year is the beginning book value minus the depreciation for the year. Remember, to find the amount of depreciation for any year, you multiply the purchase price of the asset times the MACRS percentage for the year. The depreciation schedule for this asset is:
| Year | Beginning Book Value | MACRS | Depreciation | Ending Book value | ||
| 1 | $847,000.00 | 0.1429 | $121,036.30 | $725,963.70 | ||
| 2 | 725,963.70 | 0.2449 | 207,430.30 | 518,533.40 | ||
| 3 | 518,533.40 | 0.1749 | 148,140.30 | 370,393.10 | ||
| 4 | 370,393.10 | 0.1249 | 105,790.30 | 264,602.80 | ||
| 5 | 264,602.80 | 0.0893 | 75,637.10 | 188,965.70 | ||
| 6 | 188,965.70 | 0.0893 | 75,637.10 | 113,328.60 | ||
| 7 | 113,328.60 | 0.0893 | 75,637.10 | 37,691.50 | ||
| 8 | 37,691.50 | 0.0445 | 37,691.50 | 0 |
7. The asset has an 8 year useful life and we want to find the BV of the asset after 5 years. With straight-line depreciation, the depreciation each year will be:
Annual depreciation = $440,000 / 8
Annual depreciation = $55,000
So, after five years, the accumulated depreciation will be:
Accumulated depreciation = 5($55,000)
Accumulated depreciation = $275,000
The book value at the end of year five is thus:
BV5 = $440,000 – 275,000
BV5 = $165,000
The asset is sold at a loss to book value, so the depreciation tax shield of the loss is recaptured.
Aftertax salvage value = $55,000 + ($165,000 – 55,000)(0.35)
Aftertax salvage value = $93,500
To find the taxes on salvage value, remember to use the equation:
Taxes on salvage value = (BV – MV)tc
This equation will always give the correct sign for a tax inflow (refund) or outflow (payment).
8. To find the BV at the end of four years, we need to find the accumulated depreciation for the first four years. We could calculate a table as in Problem 6, but an easier way is to add the MACRS depreciation amounts for each of the first four years and multiply this percentage times the cost of the asset. We can then subtract this from the asset cost. Doing so, we get:
BV4 = $9.3M – 9.3M(0.2000 + 0.3200 + 0.1920 + 0.1152)
BV4 = $1,607,040
The asset is sold at a gain to book value, so this gain is taxable.
Aftertax salvage value = $2,100,000 + ($1,607,040 – 2,100,000)(.35)
Aftertax salvage value = $1,927,464
9. Using the tax shield approach to calculating OCF (Remember the approach is irrelevant; the final answer will be the same no matter which of the four methods you use.), we get:
OCF = (Sales – Costs)(1 – tC) + tCDepreciation
OCF = ($2.4M – 960K)(1 – 0.35) + 0.35($2.7M/3)
OCF = $1,251,000
10. Since we have the OCF, we can find the NPV as the initial cash outlay, plus the PV of the OCFs, which are an annuity, so the NPV is:
NPV = –$2.7M + $1,251,000(PVIFA15%,3)
NPV = $156,314.62
11. The cash outflow at the beginning of the project will increase because of the spending on NWC. At the end of the project, the company will recover the NWC, so it will be a cash inflow. The sale of the equipment will result in a cash inflow, but we also must account for the taxes which will be paid on this sale. So, the cash flows for each year of the project will be:
| Year | Cash Flow | |||
| 0 | – $3,000,000 | = –$2.7M – 300K | ||
| 1 | 1,251,000 | |||
| 2 | 1,251,000 | |||
| 3 | 1,687,500 | = $1,251,000 + 300,000 + 210,000 + (0 – 210,000)(.35) |
And the NPV of the project is:
NPV = –$3,000,000 + $1,251,000,500(PVIFA15%,2) + ($1,687,500 / 1.153)
NPV = $143,320.46
12. First we will calculate the annual depreciation for the equipment necessary for the project. The depreciation amount each year will be:
Year 1 depreciation = $2.7M(0.3333) = $899,910
Year 2 depreciation = $2.7M(0.4444) = $1,199,880
Year 3 depreciation = $2.7M(0.1482) = $400,140
So, the book value of the equipment at the end of three years, which will be the initial investment minus the accumulated depreciation, is:
Book value in 3 years = $2.7M – ($899,910 + 1,199,880 + 400,140)
Book value in 3 years = $200,070
The asset is sold at a gain to book value, so this gain is taxable.
Aftertax salvage value = $210,000 + ($200,070 – 210,000)(0.35)
Aftertax salvage value = $206,525
To calculate the OCF, we will use the tax shield approach, so the cash flow each year is:
OCF = (Sales – Costs)(1 – tC) + tCDepreciation
| Year | Cash Flow | |||
| 0 | – $3,000,000 | = –$2.7M – 300K | ||
| 1 | 1,250,968.50 | = ($1,440,000)(.65) + 0.35($899,910) | ||
| 2 | 1,355,958.00 | = ($1,440,000)(.65) + 0.35($1,199,880) | ||
| 3 | 1,582,573.50 | = ($1,440,000)(.65) + 0.35($400,140) + $206,525 + 300,000 |
Remember to include the NWC cost in Year 0, and the recovery of the NWC at the end of the project. The NPV of the project with these assumptions is:
NPV = – $3.0M + ($1,250,968.50/1.15) + ($1,355,958/1.152) + ($1,582,573.50/1.153)
NPV = $153,665.52
13. First we will calculate the annual depreciation of the new equipment. It will be:
Annual depreciation = $390,000/5
Annual depreciation = $78,000
Now, we calculate the aftertax salvage value. The aftertax salvage value is the market price minus (or plus) the taxes on the sale of the equipment, so:
Aftertax salvage value = MV + (BV – MV)tc
Very often the book value of the equipment is zero as it is in this case. If the book value is zero, the equation for the aftertax salvage value becomes:
Aftertax salvage value = MV + (0 – MV)tc
Aftertax salvage value = MV(1 – tc)
We will use this equation to find the aftertax salvage value since we know the book value is zero. So, the aftertax salvage value is:
Aftertax salvage value = $60,000(1 – 0.34)
Aftertax salvage value = $39,600
Using the tax shield approach, we find the OCF for the project is:
OCF = $120,000(1 – 0.34) + 0.34($78,000)
OCF = $105,720
Now we can find the project NPV. Notice we include the NWC in the initial cash outlay. The recovery of the NWC occurs in Year 5, along with the aftertax salvage value.
NPV = –$390,000 – 28,000 + $105,720(PVIFA10%,5) + [($39,600 + 28,000) / 1.15]
NPV = $24,736.26
14. First we will calculate the annual depreciation of the new equipment. It will be:
Annual depreciation charge = $925,000/5
Annual depreciation charge = $185,000
The aftertax salvage value of the equipment is:
Aftertax salvage value = $90,000(1 – 0.35)
Aftertax salvage value = $58,500
Using the tax shield approach, the OCF is:
OCF = $360,000(1 – 0.35) + 0.35($185,000)
OCF = $298,750
Now we can find the project IRR. There is an unusual feature that is a part of this project. Accepting this project means that we will reduce NWC. This reduction in NWC is a cash inflow at Year 0. This reduction in NWC implies that when the project ends, we will have to increase NWC. So, at the end of the project, we will have a cash outflow to restore the NWC to its level before the project. We also must include the aftertax salvage value at the end of the project. The IRR of the project is:
NPV = 0 = –$925,000 + 125,000 + $298,750(PVIFAIRR%,5) + [($58,500 – 125,000) / (1+IRR)5]
IRR = 23.85%
15. To evaluate the project with a $400,000 cost savings, we need the OCF to compute the NPV. Using the tax shield approach, the OCF is:
OCF = $400,000(1 – 0.35) + 0.35($185,000) = $324,750
NPV = – $925,000 + 125,000 + $324,750(PVIFA20%,5) + [($58,500 – 125,000) / (1.20)5]
NPV = $144,476.43
The NPV with a $300,000 cost savings is:
OCF = $300,000(1 – 0.35) + 0.35($185,000)
OCF = $259,750
NPV = – $925,000 + 125,000 + $259,750(PVIFA20%,5) + [($58,500 – 125,000) / (1.20)5]
NPV = – $49,913.36
We would accept the project if cost savings were $400,000, and reject the project if the cost savings were $300,000. The required pretax cost savings that would make us indifferent about the project is the cost savings that results in a zero NPV. The NPV of the project is:
NPV = 0 = – $925,000 + $125,000 + OCF(PVIFA20%,5) + [($58,500 – 125,000) / (1.20)5]
Solving for the OCF, we find the necessary OCF for zero NPV is:
OCF = $276,440.01
Using the tax shield approach to calculating OCF, we get:
OCF = $276,440.01 = (S – C)(1 – 0.35) + 0.35($185K)
(S – C) = $325,676.94
The cost savings that will make us indifferent is $325,676.94.
16. To calculate the EAC of the project, we first need the NPV of the project. Notice that we include the NWC expenditure at the beginning of the project, and recover the NWC at the end of the project. The NPV of the project is:
NPV = – $210,000 – 20,000 – $32,000(PVIFA15%,5) + $20,000/1.155 = –$327,325.43
Now we can find the EAC of the project. The EAC is:
EAC = – $327,325.43 / (PVIFA15%,5) = –$97,646.27
17. We will need the aftertax salvage value of the equipment to compute the EAC. Even though the equipment for each product has a different initial cost, both have the same salvage value. The aftertax salvage value for both is:
Both cases: aftertax salvage value = $20,000(1 – 0.35) = $13,000
To calculate the EAC, we first need the OCF and NPV of each option. The OCF and NPV for Techron I is:
OCF = – $34,000(1 – 0.35) + 0.35($210,000/3) = $2,400
NPV = –$210,000 + $2,400(PVIFA14%,3) + ($13,000/1.143) = –$195,653.45
EAC = –$195,653.45 / (PVIFA14%,3) = –$84,274.10
And the OCF and NPV for Techron II is:
OCF = – $23,000(1 – 0.35) + 0.35($320,000/5) = $7,450
NPV = –$320,000 + $7,450(PVIFA14%,5) + ($13,000/1.145) = –$287,671.75
EAC = –$287,671.75 / (PVIFA14%,5) = –$83,794.05
The two milling machines have unequal lives, so they can only be compared by expressing both on an equivalent annual basis, which is what the EAC method does. Thus, you prefer the Techron II because it has the lower (less negative) annual cost.
18. To find the bid price, we need to calculate all other cash flows for the project, and then solve for the bid price. The aftertax salvage value of the equipment is:
Aftertax salvage value = $50,000(1 – 0.35) = $32,500
Now we can solve for the necessary OCF that will give the project a zero NPV. The equation for the NPV of the project is:
NPV = 0 = – $780,000 – 75,000 + OCF(PVIFA16%,5) + [($75,000 + 32,500) / 1.165]
Solving for the OCF, we find the OCF that makes the project NPV equal to zero is:
OCF = $803,817.85 / PVIFA16%,5 = $245,493.51
The easiest way to calculate the bid price is the tax shield approach, so:
OCF = $245,493.51 = [(P – v)Q – FC ](1 – tc) + tcD
$245,493.51 = [(P – $8.50)(150,000) – $240,000 ](1 – 0.35) + 0.35($780,000/5)
P = $12.06
Intermediate
19. First, we will calculate the depreciation each year, which will be:
D1 = $480,000(0.2000) = $96,000
D2 = $480,000(0.3200) = $153,600
D3 = $480,000(0.1920) = $92,160
D4 = $480,000(0.1152) = $55,296
The book value of the equipment at the end of the project is:
BV4 = $480,000 – ($96,000 + 153,600 + 92,160 + 55,296) = $82,944
The asset is sold at a loss to book value, so this creates a tax refund.
After-tax salvage value = $70,000 + ($82,944 – 70,000)(0.35) = $74,530.40
So, the OCF for each year will be:
OCF1 = $160,000(1 – 0.35) + 0.35($96,000) = $137,600.00
OCF2 = $160,000(1 – 0.35) + 0.35($153,600) = $157,760.00
OCF3 = $160,000(1 – 0.35) + 0.35($92,160) = $136,256.00
OCF4 = $160,000(1 – 0.35) + 0.35($55,296) = $123,353.60
Now we have all the necessary information to calculate the project NPV. We need to be careful with the NWC in this project. Notice the project requires $20,000 of NWC at the beginning, and $3,000 more in NWC each successive year. We will subtract the $20,000 from the initial cash flow, and subtract $3,000 each year from the OCF to account for this spending. In Year 4, we will add back the total spent on NWC, which is $29,000. The $3,000 spent on NWC capital during Year 4 is irrelevant. Why? Well, during this year the project required an additional $3,000, but we would get the money back immediately. So, the net cash flow for additional NWC would be zero. With all this, the equation for the NPV of the project is:
NPV = – $480,000 – 20,000 + ($137,600 – 3,000)/1.14 + ($157,760 – 3,000)/1.142
+ ($136,256 – 3,000)/1.143 + ($123,353.60 + 29,000 + 74,530.40)/1.144
NPV = –$38,569.48
20. If we are trying to decide between two projects that will not be replaced when they wear out, the proper capital budgeting method to use is NPV. Both projects only have costs associated with them, not sales, so we will use these to calculate the NPV of each project. Using the tax shield approach to calculate the OCF, the NPV of System A is:
OCFA = –$120,000(1 – 0.34) + 0.34($430,000/4)
OCFA = –$42,650
NPVA = –$430,000 – $42,650(PVIFA20%,4)
NPVA = –$540,409.53
And the NPV of System B is:
OCFB = –$80,000(1 – 0.34) + 0.34($540,000/6)
OCFB = –$22,200
NPVB = –$540,000 – $22,200(PVIFA20%,6)
NPVB = –$613,826.32
If the system will not be replaced when it wears out, then System A should be chosen, because it has the more positive NPV.
21. If the equipment will be replaced at the end of its useful life, the correct capital budgeting technique is EAC. Using the NPVs we calculated in the previous problem, the EAC for each system is:
EACA = – $540,409.53 / (PVIFA20%,4)
EACA = –$208,754.32
EACB = – $613,826.32 / (PVIFA20%,6)
EACB = –$184,581.10
If the conveyor belt system will be continually replaced, we should choose System B since it has the more positive NPV.
22. To find the bid price, we need to calculate all other cash flows for the project, and then solve for the bid price. The aftertax salvage value of the equipment is:
After-tax salvage value = $600,000(1 – 0.34)
After-tax salvage value = $396,000
Now we can solve for the necessary OCF that will give the project a zero NPV. The equation for the NPV of the project is:
NPV = 0 = – $3,100,000 – 1,200,000 – 600,000 + OCF (PVIFA15%,5) – $50,000(PVIFA15%,4)
+ {($396,000 + 600,000 + 4(50,000)] / 1.155}
Solving for the OCF, we find the OCF that makes the project NPV equal to zero is:
OCF = $4,448,125.54 / PVIFA15%,5
OCF = $1,326,945.03
The easiest way to calculate the bid price is the tax shield approach, so:
OCF = $1,326,945.03 = [(P – v)Q – FC ](1 – tC) + tcD
$1,326,945.03 = [(P – $0.005)(80,000,000) – $800,000](1 – 0.34) + 0.34($3,100,000/5)
P = $0.03614
23. At a given price, taking accelerated depreciation compared to straight-line depreciation causes the NPV to be higher; similarly, at a given price, lower net working capital investment requirements will cause the NPV to be higher. Thus, NPV would be zero at a lower price in this situation. In the case of a bid price, you could submit a lower price and still break-even, or submit the higher price and make a positive NPV.
24. Since we need to calculate the EAC for each machine, sales are irrelevant. EAC only uses the costs of operating the equipment, not the sales. Using the bottom up approach, or net income plus depreciation, method to calculate OCF, we get:
| Machine A | Machine B | |||
| Variable costs | –$3,150,000 | –$2,700,000 | ||
| Fixed costs | –150,000 | –100,000 | ||
| Depreciation | –350,000 | –500,000 | ||
| EBT | –$3,650,000 | –$3,300,000 | ||
| Tax | 1,277,500 | 1,155,000 | ||
| Net income | –$2,372,500 | –$2,145,000 | ||
| + Depreciation | 350,000 | 500,000 | ||
| OCF | –$2,022,500 | –$1,645,000 |
The NPV and EAC for Machine A is:
NPVA = –$2,100,000 – $2,022,500(PVIFA10%,6)
NPVA = –$10,908,514.76
EACA = – $10,908,514.76 / (PVIFA10%,6)
EACA = –$2,504,675.50
And the NPV and EAC for Machine B is:
NPVB = –$4,500,000 – 1,645,000(PVIFA10%,9)
NPVB = –$13,973,594.18
EACB = – $13,973,594.18 / (PVIFA10%,9)
EACB = –$2,426,382.43
You should choose Machine B since it has a more positive EAC.
Challenge
25. This is an in-depth capital budgeting problem. Probably the easiest OCF calculation for this problem is the bottom up approach, so we will construct an income statement for each year. Beginning with the initial cash flow at time zero, the project will require an investment in equipment. The project will also require an investment in NWC. The NWC investment will be 15 percent of the next year’s sales. In this case, it will be Year 1 sales. Realizing we need Year 1 sales to calculate the required NWC capital at time 0, we find that Year 1 sales will be $27,625,000. So, the cash flow required for the project today will be:
| Capital spending | –$21,000,000 | |
| Change in NWC | –1,500,000 | |
| Total cash flow | –$22,500,000 |
Now we can begin the remaining calculations. Sales figures are given for each year, along with the price per unit. The variable costs per unit are used to calculate total variable costs, and fixed costs are given at $900,000 per year. To calculate depreciation each year, we use the initial equipment cost of $21 million, times the appropriate MACRS depreciation each year. The remainder of each income statement is calculated below. Notice at the bottom of the income statement we added back depreciation to get the OCF for each year. The section labeled “Net cash flows” will be discussed below:
| Year | 1 | 2 | 3 | 4 | 5 | |
| Ending book value | $ 17,999,100 | $ 12,856,200 | $ 9,183,300 | $ 6,560,400 | $ 4,685,100 | |
| Sales | $ 27,625,000 | $ 31,850,000 | $ 34,450,000 | $ 37,050,000 | $ 30,225,000 | |
| Variable costs | 20,400,000 | 23,520,000 | 25,440,000 | 27,360,000 | 22,320,000 | |
| Fixed costs | 900,000 | 900,000 | 900,000 | 900,000 | 900,000 | |
| Depreciation | 3,000,900 | 5,142,900 | 3,672,900 | 2,622,900 | 1,875,300 | |
| EBIT | 3,324,100 | 2,287,100 | 4,437,100 | 6,167,100 | 5,129,700 | |
| Taxes | 1,163,435 | 800,485 | 1,552,985 | 2,158,485 | 1,795,395 | |
| Net income | 2,160,665 | 1,486,615 | 2,884,115 | 4,008,615 | 3,334,305 | |
| Depreciation | 3,000,900 | 5,142,900 | 3,672,900 | 2,622,900 | 1,875,300 | |
| Operating cash flow | $ 5,161,565 | $ 6,629,515 | $ 6,557,015 | $ 6,631,515 | $ 5,209,605 | |
| Net cash flows | ||||||
| Operating cash flow | $ 5,161,565 | $ 6,629,515 | $ 6,557,015 | $ 6,631,515 | $ 5,209,605 | |
| Change in NWC | –633,750 | –390,000 | –390,000 | 1,023,750 | 1,890,000 | |
| Capital spending | 0 | 0 | 0 | 0 | 4,369,785 | |
| Total cash flow | $ 4,527,815 | $ 6,239,515 | $ 6,167,015 | $ 7,655,265 | $ 11,469,390 |
After we calculate the OCF for each year, we need to account for any other cash flows. The other cash flows in this case are NWC cash flows and capital spending, which is the aftertax salvage of the equipment. The required NWC capital is 15 percent of the sales in the next year. We will work through the NWC cash flow for Year 1. The total NWC in Year 1 will be 15 percent of sales increase from Year 1 to Year 2, or:
Increase in NWC for Year 1 = .15($31,850,000 – 27,625,000)
Increase in NWC for Year 1 = $633,750
Notice that the NWC cash flow is negative. Since the sales are increasing, we will have to spend more money to increase NWC. In Year 4, the NWC cash flow is positive since sales are declining. And, in Year 5, the NWC cash flow is the recovery of all NWC the company still has in the project.
To calculate the aftertax salvage value, we first need the book value of the equipment. The book value at the end of the five years will be the purchase price, minus the total depreciation. So, the ending book value is:
Ending book value = $21,000,000 – ($3,000,900 + 5,142,900 + 3,672,900 + 2,622,900 + 1,875,300)
Ending book value = $4,685,100
The market value of the used equipment is 20 percent of the purchase price, or $4.2 million, so the aftertax salvage value will be:
Aftertax salvage value = $4,200,000 + ($4,685,100 – 4,200,000)(.35)
Aftertax salvage value = $4,369,785
The aftertax salvage value is included in the total cash flows are capital spending. Now we have all of the cash flows for the project. The NPV of the project is:
NPV = –$22,500,000 + $4,527,815/1.18 + $6,239,515/1.182 + $6,167,015/1.183 + $7,655,265/1.184
+ $11,469,390/1.185
NPV = –$1,466,433.80
And the IRR is:
NPV = 0 = –$22,500,000 + $4,527,815/(1 + IRR) + $6,239,515/(1 + IRR)2 + $6,167,015/(1 + IRR)3
+ 7$,655,265/(1 + IRR)4 + $11,469,390/(1 + IRR)5
IRR = 15.47%
We should reject the project.
26. To find the initial pretax cost savings necessary to buy the new machine, we should use the tax shield approach to find the OCF. We begin by calculating the depreciation each year using the MACRS depreciation schedule. The depreciation each year is:
D1 = $480,000(0.3333) = $159,984
D2 = $480,000(0.4444) = $213,312
D3 = $480,000(0.1482) = $71,136
D4 = $480,000(0.0741) = $35,568
Using the tax shield approach, the OCF each year is:
OCF1 = (S – C)(1 – 0.35) + 0.35($159,984)
OCF2 = (S – C)(1 – 0.35) + 0.35($213,312)
OCF3 = (S – C)(1 – 0.35) + 0.35($71,136)
OCF4 = (S – C)(1 – 0.35) + 0.35($35,568)
OCF5 = (S – C)(1 – 0.35)
Now we need the aftertax salvage value of the equipment. The aftertax salvage value is:
After-tax salvage value = $45,000(1 – 0.35) = $29,250
To find the necessary cost reduction, we must realize that we can split the cash flows each year. The OCF in any given year is the cost reduction (S – C) times one minus the tax rate, which is an annuity for the project life, and the depreciation tax shield. To calculate the necessary cost reduction, we would require a zero NPV. The equation for the NPV of the project is:
NPV = 0 = – $480,000 – 40,000 + (S – C)(0.65)(PVIFA12%,5) + 0.35($159,984/1.12
+ $213,312/1.122 + $71,136/1.123 + $35,568/1.124) + ($40,000 + 29,250)/1.125
Solving this equation for the sales minus costs, we get:
(S – C)(0.65)(PVIFA12%,5) = $345,559.78
(S – C) = $147,479.45
27. a . This problem is basically the same as Problem 18, except we are given a sales price. The cash flow at Time 0 for all three parts of this question will be:
| Capital spending | –$780,000 | |
| Change in NWC | –75,000 | |
| Total cash flow | –$855,000 |
We will use the initial cash flow and the salvage value we already found in that problem. Using the bottom up approach to calculating the OCF, we get:
Assume price per unit = $13 and units/year = 150,000
| Year | 1 | 2 | 3 | 4 | 5 | |
| Sales | $1,950,000 | $1,950,000 | $1,950,000 | $1,950,000 | $1,950,000 | |
| Variable costs | 1,275,000 | 1,275,000 | 1,275,000 | 1,275,000 | 1,275,000 | |
| Fixed costs | 240,000 | 240,000 | 240,000 | 240,000 | 240,000 | |
| Depreciation | 156,000 | 156,000 | 156,000 | 156,000 | 156,000 | |
| EBIT | 279,000 | 279,000 | 279,000 | 279,000 | 279,000 | |
| Taxes (35%) | 97,650 | 97,650 | 97,650 | 97,650 | 97,650 | |
| Net Income | 181,350 | 181,350 | 181,350 | 181,350 | 181,350 | |
| Depreciation | 156,000 | 156,000 | 156,000 | 156,000 | 156,000 | |
| Operating CF | $337,350 | $337,350 | $337,350 | $337,350 | $337,350 |
| Year | 1 | 2 | 3 | 4 | 5 | |
| Operating CF | $337,350 | $337,350 | $337,350 | $337,350 | $337,350 | |
| Change in NWC | 0 | 0 | 0 | 0 | 75,000 | |
| Capital spending | 0 | 0 | 0 | 0 | 32,500 | |
| Total CF | $337,350 | $337,350 | $337,350 | $337,350 | $444,850 | |
With these cash flows, the NPV of the project is:
NPV = – $780,000 – 75,000 + $337,350(PVIFA16%,5) + [($75,000 + 32,500) / 1.165]
NPV = $300,765.11
If the actual price is above the bid price that results in a zero NPV, the project will have a positive NPV. As for the cartons sold, if the number of cartons sold increases, the NPV will increase, and if the costs increase, the NPV will decrease.
b. To find the minimum number of cartons sold to still breakeven, we need to use the tax shield approach to calculating OCF, and solve the problem similar to finding a bid price. Using the initial cash flow and salvage value we already calculated, the equation for a zero NPV of the project is:
NPV = 0 = – $780,000 – 75,000 + OCF(PVIFA16%,5) + [($75,000 + 32,500) / 1.165]
So, the necessary OCF for a zero NPV is:
OCF = $803,817.85 / PVIFA16%,5 = $245,493.51
Now we can use the tax shield approach to solve for the minimum quantity as follows:
OCF = $245,493.51 = [(P – v)Q – FC ](1 – tc) + tcD
$245,493.51 = [($13.00 – 8.50)Q – 240,000 ](1 – 0.35) + 0.35($780,000/5)
Q = 118,596
As a check, we can calculate the NPV of the project with this quantity. The calculations are:
| Year | 1 | 2 | 3 | 4 | 5 | |
| Sales | $1,541,748 | $1,541,748 | $1,541,748 | $1,541,748 | $1,541,748 | |
| Variable costs | 1,008,066 | 1,008,066 | 1,008,066 | 1,008,066 | 1,008,066 | |
| Fixed costs | 240,000 | 240,000 | 240,000 | 240,000 | 240,000 | |
| Depreciation | 156,000 | 156,000 | 156,000 | 156,000 | 156,000 | |
| EBIT | 137,682 | 137,682 | 137,682 | 137,682 | 137,682 | |
| Taxes (35%) | 48,189 | 48,189 | 48,189 | 48,189 | 48,189 | |
| Net Income | 89,493 | 89,493 | 89,493 | 89,493 | 89,493 | |
| Depreciation | 156,000 | 156,000 | 156,000 | 156,000 | 156,000 | |
| Operating CF | $245,493 | $245,493 | $245,493 | $245,493 | $245,493 |
| Year | 1 | 2 | 3 | 4 | 5 | |
| Operating CF | $245,493 | $245,493 | $245,493 | $245,493 | $245,493 | |
| Change in NWC | 0 | 0 | 0 | 0 | 75,000 | |
| Capital spending | 0 | 0 | 0 | 0 | 32,500 | |
| Total CF | $245,493 | $245,493 | $245,493 | $245,493 | $352,993 | |
NPV = – $780,000 – 75,000 + $245,493(PVIFA16%,5) + [($75,000 + 32,500) / 1.165] » $0
Note, the NPV is not exactly equal to zero because we had to round the number of cartons sold, you cannot sell one-half of a carton.
c. To find the highest level of fixed costs and still breakeven, we need to use the tax shield approach to calculating OCF, and solve the problem similar to finding a bid price. Using the initial cash flow and salvage value we already calculated, the equation for a zero NPV of the project is:
NPV = 0 = – $780,000 – 75,000 + OCF(PVIFA16%,5) + [($75,000 + 32,500) / 1.165]
OCF = $803,817.85 / PVIFA16%,5 = $245,493.51
Notice this is the same OCF we calculated in part b. Now we can use the tax shield approach to solve for the maximum level of fixed costs as follows:
OCF = $245,493.51 = [(P–v)Q – FC ](1 – tC) + tCD
$245,493.51 = [($13.00 – $8.50)(150,000) – FC](1 – 0.35) + 0.35($780,000/5)
FC = $381,317.67
As a check, we can calculate the NPV of the project with this quantity. The calculations are:
| Year | 1 | 2 | 3 | 4 | 5 | |
| Sales | $1,950,000 | $1,950,000 | $1,950,000 | $1,950,000 | $1,950,000 | |
| Variable costs | 1,275,000 | 1,275,000 | 1,275,000 | 1,275,000 | 1,275,000 | |
| Fixed costs | 381,318 | 381,318 | 381,318 | 381,318 | 381,318 | |
| Depreciation | 156,000 | 156,000 | 156,000 | 156,000 | 156,000 | |
| EBIT | 137,682 | 137,682 | 137,682 | 137,682 | 137,682 | |
| Taxes (35%) | 48,189 | 48,189 | 48,189 | 48,189 | 48,189 | |
| Net Income | 89,494 | 89,494 | 89,494 | 89,494 | 89,494 | |
| Depreciation | 156,000 | 156,000 | 156,000 | 156,000 | 156,000 | |
| Operating CF | $245,494 | $245,494 | $245,494 | $245,494 | $245,494 |
| Year | 1 | 2 | 3 | 4 | 5 | |
| Operating CF | $245,494 | $245,494 | $245,494 | $245,494 | $245,494 | |
| Change in NWC | 0 | 0 | 0 | 0 | 75,000 | |
| Capital spending | 0 | 0 | 0 | 0 | 32,500 | |
| Total CF | $245,494 | $245,494 | $245,494 | $245,494 | $352,994 | |
NPV = – $780,000 – 75,000 + $245,493(PVIFA16%,5) + [($75,000 + 32,500) / 1.165] » $0
28. We need to find the bid price for a project, but the project has extra cash flows. Since we don’t already produce the keyboard, the sales of the keyboard outside the contract are relevant cash flows. Since we know the extra sales number and price, we can calculate the cash flows generated by these sales. The cash flow generated from the sale of the keyboard outside the contract is:
| 1 | 2 | 3 | 4 | ||
| Sales | $825,000 | $1,650,000 | $2,200,000 | $1,375,000 | |
| Variable costs | 495,000 | 990,000 | 1,320,000 | 825,000 | |
| EBT | $330,000 | $660,000 | $880,000 | $550,000 | |
| Tax | 132,000 | 264,000 | 352,000 | 220,000 | |
| Net income (and OCF) | $198,000 | $396,000 | $528,000 | $330,000 |
So, the addition to NPV of these market sales is:
NPV of market sales = $198,000/1.13 + $396,000/1.132 + $528,000/1.133 + $330,000/1.134
NPV of market sales = $1,053,672.99
You may have noticed that we did not include the initial cash outlay, depreciation or fixed costs in the calculation of cash flows from the market sales. The reason is that it is irrelevant whether or not we include these here. Remember, we are not only trying to determine the bid price, but we are also determining whether or not the project is feasible. In other words, we are trying to calculate the NPV of the project, not just the NPV of the bid price. We will include these cash flows in the bid price calculation. The reason we stated earlier that whether we included these costs in this initial calculation was irrelevant is that you will come up with the same bid price if you include these costs in this calculation, or if you include them in the bid price calculation.
Next, we need to calculate the aftertax salvage value, which is:
Aftertax salvage value = $200,000(1 – .40) = $120,000
Instead of solving for a zero NPV as is usual in setting a bid price, the company president requires an NPV of $100,000, so we will solve for a NPV of that amount. The NPV equation for this project is (remember to include the NWC cash flow at the beginning of the project, and the NWC recovery at the end):
NPV = $100,000 = –$2,400,000 – 75,000 + 1,053,672.99 + OCF (PVIFA13%,4) + [($120,000 +
75,000) / 1.134]
Solving for the OCF, we get:
OCF = $1,401,729.86 / PVIFA13%,4 = $471,253.44
Now we can solve for the bid price as follows:
OCF = $471,253.44 = [(P – v)Q – FC ](1 – tC) + tCD
$471,253.44 = [(P – $165)(10,000) – $500,000](1 – 0.40) + 0.40($2,400,000/4)
P = $253.54
29. Since the two computers have unequal lives, the correct method to analyze the decision is the EAC. We will begin with the EAC of the new computer. Using the depreciation tax shield approach, the OCF for the new computer system is:
OCF = ($125,000)(1 – .38) + ($780,000 / 5)(.38) = $136,780
Notice that the costs are positive, which represents a cash inflow. The costs are positive in this case since the new computer will generate a cost savings. The only initial cash flow for the new computer is cost of $780,000. We next need to calculate the aftertax salvage value, which is:
Aftertax salvage value = $140,000(1 – .38) = $86,800
Now we can calculate the NPV of the new computer as:
NPV = –$780,000 + $136,780(PVIFA14%,5) + $86,800 / 1.145
NPV = –$265,341.99
And the EAC of the new computer is:
EAC = – $265,341.99 / (PVIFA14%,5) = –$77,289.75
Analyzing the old computer, the only OCF is the depreciation tax shield, so:
OCF = $130,000(.38) = $49,400
The initial cost of the old computer is a little trickier. You might assume that since we already own the old computer there is no initial cost, but we can sell the old computer, so there is an opportunity cost. We need to account for this opportunity cost. To do so, we will calculate the aftertax salvage value of the old computer today. We need the book value of the old computer to do so. The book value is not given directly, but we are told that the old computer has depreciation of $130,000 per year for the next three years, so we can assume the book value is the total amount of depreciation over the remaining life of the system, or $390,000. So, the aftertax salvage value of the old computer is:
Aftertax salvage value = $230,000 + ($390,000 – 230,000)(.38) = $290,800
This is the initial cost of the old computer system today because we are forgoing the opportunity to sell it today. We next need to calculate the aftertax salvage value of the computer system in two years since we are “buying” it today. The aftertax salvage value in two years is:
Aftertax salvage value = $90,000 + ($130,000 – 90,000)(.38) = $105,200
Now we can calculate the NPV of the old computer as:
NPV = –$290,800 + $49,400(PVIFA14%,2) + 105,200 / 1.142
NPV = –$128,506.99
And the EAC of the old computer is:
EAC = – $128,506.99 / (PVIFA14%,2) = –$78,040.97
If we are going to replace the system in two years no matter what our decision today, we should instead replace it today since the EAC is lower.
b. If we are only concerned with whether or not to replace the machine now, and are not worrying about what will happen in two years, the correct analysis is NPV. To calculate the NPV of the decision on the computer system now, we need the difference in the total cash flows of the old computer system and the new computer system. From our previous calculations, we can say the cash flows for each computer system are:
| t | New computer | Old computer | Difference | |
| 0 | –$780,000 | $290,800 | –$489,200 | |
| 1 | 136,780 | –49,400 | 87,380 | |
| 2 | 136,780 | –154,600 | –17,820 | |
| 3 | 136,780 | 0 | 136,780 | |
| 4 | 136,780 | 0 | 136,780 | |
| 5 | 223,580 | 0 | 223,580 |
Since we are only concerned with marginal cash flows, the cash flows of the decision to replace the old computer system with the new computer system are the differential cash flows. The NPV of the decision to replace, ignoring what will happen in two years is:
NPV = –$489,200 + $87,380/1.14 – $17,820/1.142 + $136,780/1.143 + $136,780/1.144
+ $223,580/1.145
NPV = –$136,835.00
If we are not concerned with what will happen in two years, we should not replace the old computer system.
