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Year 4
Year 5
Year 6
Year 7
Revenues
$538,095.24
$538,095.24
$538,095.24
$538,095.24
Costs
319,272.80
316,232.10
313,220.37
310,237.32
Depreciation
76,393.80
72,756.00
69,291.43
65,991.84
EBT
$142,428.64
$149,107.13
$155,583.44
$161,866.08
We can also find the NPV using real cash flows and the nominal required return. This will allow us to
find the operating cash flow using the tax shield approach. Both the revenues and expenses are growing
annuities, but growing at different rates. This means we must find the present value of each separately.
We also need to account for the effect of taxes, so we will multiply by one minus the tax rate. So, the
present value of the aftertax revenues using the growing annuity equation is:
first year is a nominal value, so we can find the present value of the depreciation tax shield as an
ordinary annuity using the nominal required return. So, the present value of the depreciation tax shield
will be:
PV of depreciation tax shield = ($650,000/7)(.24)(PVIFA12.35%,7)
PV of depreciation tax shield = $100,587.68
27. Here we have a project in which the quantity sold each year increases. First, we need to calculate the
quantity sold each year by increasing the current year’s quantity by the growth rate. So, the quantity
sold each year will be:
Year 1 quantity = 10,400
Year 2 quantity = 10,400(1 + .08) = 11,232
Now we can calculate the sales revenue and variable costs each year. The pro forma income statements
and operating cash flow each year will be:
Year 0
Year 1
Year 2
Year 3
Year 4
Year 5
Revenues
$634,400.00
$685,152.00
$739,964.16
$799,161.29
$863,094.20
Fixed costs
125,000.00
125,000.00
125,000.00
125,000.00
125,000.00
So, the NPV of the project is:
NPV = –$620,000 + $270,472/1.18 + $298,077.76/1.182 + $327,891.98/1.183
+ $360,091.34/1.184 + $439,866.65/1.185
We could also calculate the cash flows using the tax shield approach, with growing annuities and
ordinary annuities. The sales and variable costs increase at the same rate as sales, so both are growing
annuities. The fixed costs and depreciation are both ordinary annuities. Using the growing annuity
equation, the present value of the revenues is:
And the present value of the variable costs will be:
PV of variable costs = $706,900.48
The fixed costs and depreciation are both ordinary annuities. The present value of each is:
PV of fixed costs = C({1 – [1/(1 + r)]t}/r)
PV of fixed costs = $125,000(PVIFA18%,5)
Now, we can use the depreciation tax shield approach to find the NPV of the project, which is:
28. We will begin by calculating the aftertax salvage value of the equipment at the end of the project’s
life. The aftertax salvage value is the market value of the equipment minus any taxes paid (or
refunded), so the aftertax salvage value in four years will be:
Taxes on salvage value = (BV – MV)tC
Taxes on salvage value = ($0 – 310,000)(.23)
NWC
–225,000
225,000
Total cash flow
–$4,525,000
$1,389,210
$1,635,867
$1,663,591
$2,650,593
Notice the calculation of the cash flow at Time 0. The capital spending on equipment and investment
29. The aftertax salvage value will be the same as the previous problem, $238,700. We need to calculate
the operating cash flow each year. Note, we assume that the net working capital cash flow occurs
immediately. Using the bottom up approach to calculating operating cash flow, we find:
Year 0
Year 1
Year 2
Year 3
Year 4
Revenues
$2,165,500
$2,409,500
$2,806,000
$1,860,500
Fixed costs
375,000
375,000
375,000
375,000
Capital spending
–$3,400,000
238,700
Land
–900,000
1,200,000
Notice the calculation of the cash flow at Time 0. The capital spending on equipment and investment
30. Replacement decision analysis is the same as the analysis of two competing projects; in this case, keep
the current equipment, or purchase the new equipment. We will consider the purchase of the new
machine first.
Purchase new machine:
The initial cash outlay for the new machine is the cost of the new machine. We can calculate the
end of five years, so we need to include this in the cash flows analysis. The aftertax salvage value will
be:
Sell machine
$900,000
Taxes
–189,000
Total
$711,000
The NPV of purchasing the new machine is:
could sell the old machine. Also, if the company sells the old machine at its current value, it will incur
taxes. Both of these cash flows need to be included in the analysis. So, the initial cash flow of keeping
the old machine will be:
Keep machine
–$2,800,000
Taxes
252,000
Next, we can calculate the operating cash flow created if the company keeps the old machine. We need
to account for the cost of maintenance, as well as the cash flow effects of depreciation. The pro forma
income statement, adding depreciation to net income to calculate the operating cash flow will be:
Maintenance cost
$855,000
Depreciation
320,000
EBT
–$1,175,000
The old machine also has a salvage value at the end of five years, so we need to include this in the
cash flows analysis. The aftertax salvage value will be:
Sell machine
$140,000
Taxes
–29,400
The company should buy the new machine since it has a greater NPV.
There is another way to analyze a replacement decision that is often used. It is an incremental cash
flow analysis of the change in cash flows from the existing machine to the new machine, assuming the
new machine is purchased. In this type of analysis, the initial cash outlay would be the cost of the new
machine, and the cash inflow (including any applicable taxes) of selling the old machine. In this case,
the initial cash flow under this method would be:
Purchase new machine
–$4,500,000
The cash flows from purchasing the new machine would be the difference in the operating expenses.
We would also need to include the change in depreciation. The old machine has a depreciation of
the new machine, and incur taxes on the sale in five years. However, we must also include the lost sale
of the old machine. Since we assumed we sold the old machine in the initial cash outlay, we lose the
ability to sell the machine in five years. This is an opportunity loss that must be accounted for. So, the
salvage value is:
Sell machine
$900,000
31. The project has a sales price that increases at 3 percent per year, and a variable cost per unit that
increases at 4 percent per year. First, we need to find the sales price and variable cost for each year.
The table below shows the price per unit and the variable cost per unit each year.
Year 1
Year 2
Year 3
Year 4
Year 5
Using the sales price and variable cost, we can now construct the pro forma income statement for each
year. We can use this income statement to calculate the cash flow each year. We must also make sure
to include the net working capital outlay at the beginning of the project, and the recovery of the net
working capital at the end of the project. The pro forma income statement and cash flows for each year
will be:
Year 0
Year 1
Year 2
Year 3
Year 4
Year 5
Revenues
$1,175,000.00
$1,210,250.00
$1,246,557.50
$1,283,954.23
$1,322,472.85
Fixed costs
235,000.00
235,000.00
235,000.00
235,000.00
235,000.00
Cap. spend.
–$1,200,000
With these cash flows, the NPV of the project is:
NPV = –$1,400,000 + $457,250/1.11 + $471,667.50/1.112 + $486,383.23/1.113
+ $501,400.75/1.114 + $716,723.54/1.115
The fixed costs and depreciation are both ordinary annuities. The present value of each is:
PV of fixed costs = C({1 – [1/(1 + r)]t}/r)
Challenge
32. 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 of
$1,500,000. So, the cash flow required for the project today will be:
Capital spending
–$18,500,000
Year
1
2
3
4
5
Ending book value
$15,856,350
$11,325,700
$8,090,050
$5,779,400
$4,127,350
Sales
$23,725,000
$25,675,000
$27,300,000
$26,650,000
$22,100,000
Variable costs
10,585,000
11,455,000
12,180,000
11,890,000
9,860,000
Fixed costs
3,400,000
3,400,000
3,400,000
3,400,000
3,400,000
Depreciation
2,643,650
4,530,650
3,235,650
2,310,650
1,652,050
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 is 15 percent of the sales increase 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 the sales
increase from Year 1 to Year 2, or:
The market value of the used equipment is 20 percent of the purchase price, or $3,700,000, so the
aftertax salvage value will be:
Aftertax salvage value = $3,700,000 + ($4,127,350 – 3,700,000)(.23)
Aftertax salvage value = $3,798,291
The aftertax salvage value is included in the total cash flows as capital spending. Now we have all of
the cash flows for the project. The NPV of the project is:
33. 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 = $645,000(.3333) = $214,978.50
D2 = $645,000(.4445) = $286,702.50
34. 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 = $150,000(1 – .21)
Aftertax salvage value = $118,500
35. a. This problem is basically the same as the previous problem, except that we are given a sales
price. The cash flow at Time 0 for all three parts of this question will be:
Capital spending
–$2,100,000
Change in NWC
–325,000
Total cash flow
–$2,425,000
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 = $20 and units/year = 145,000
Year
1
2
3
4
5
Sales
$2,900,000
$2,900,000
$2,900,000
$2,900,000
$2,900,000
Variable costs
1,370,250
1,370,250
1,370,250
1,370,250
1,370,250
Fixed costs
650,000
650,000
650,000
650,000
650,000
