18. The present value of the company is the present value of the future cash flows generated by the
company. Here we have real cash flows, a real interest rate, and a real growth rate. The cash flows are
a growing perpetuity, with a negative growth rate. Using the growing perpetuity equation, the present
value of the cash flows is:
PV = $5,492,753.62
19. To find the EAC, we first need to calculate the NPV of the incremental cash flows. We will begin with
the aftertax salvage value, which is:
Taxes on salvage value = (BV – MV)tC
So, the NPV of the cost of the decision to buy is:
20. We will find the EAC of the EVF first. There are no taxes since the university is tax-exempt, so the
maintenance costs are the operating cash flows. The NPV of the decision to buy one EVF is:
Since the university must buy five of the mowers, the total EAC of the decision to buy the EVF mower
is:
Total EAC = 5(–$4,710.42)
So, the EAC per mower is:
EAC = –$19,141.46/(PVIFA9%,7)
21. We will calculate the aftertax salvage value first. The aftertax salvage value of the equipment will be:
Taxes on salvage value = (BV – MV)tC
Taxes on salvage value = ($0 – 35,000)(.34)
Taxes on salvage value = –$11,900
Market price
$35,000
Tax on sale
–11,900
Aftertax salvage value
$23,100
Equipment
–$655,000
NWC
55,000
Total
–$600,000
So, the NPV of purchasing the new machine, including the recovery of the net working capital,
is:
And the IRR is:
0 = –$9,820,000 + $2,226,000(PVIFAIRR,4) + $160,000/(1 + IRR)4
Using a spreadsheet or financial calculator, we find the IRR is:
IRR = –3.10%
Now we can calculate the decision to keep the old machine:
Keep old machine:
The initial cash outlay for the new machine is the market value of the old machine, including any
potential tax. The decision to keep the old machine has an opportunity cost, namely, the company
could sell the old machine. Also, if the company sells the old machine at its current value, it will
of keeping the old machine will be:
Keep machine
–$4,370,000
Taxes
740,000
Total
–$3,630,000
Next, we can calculate the operating cash flow created if the company keeps the old machine.
There are no incremental cash flows from keeping the old machine, but we need to account for
the cash flow effects of depreciation. The income statement, adding depreciation to net income
to calculate the operating cash flow will be:
Depreciation
$630,000
EBT
–$630,000
Taxes
– 252,000
Net income
–$378,000
OCF
$252,000
So, the NPV of the decision to keep the old machine will be:
NPV = –$3,630,000 + $252,000(PVIFA10%,4)
NPV = –$2,831,193.91
And the IRR is:
0 = –$3,630,000 + $252,000(PVIFAIRR,4)
Using a spreadsheet or financial calculator, we find the IRR is:
IRR = –36.93%
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
would be:
Purchase new machine
–$9,660,000
Net working capital
–160,000
Sell old machine
4,370,000
Taxes on old machine
–740,000
Total
–$6,190,000
The cash flows from purchasing the new machine would be the saved operating expenses. We
would also need to include only the change in depreciation. The old machine has a depreciation
of $630,000 per year, and the new machine has a depreciation of $2,415,000 per year, so the
increased depreciation will be $1,785,000 per year. The pro forma income statement and
operating cash flow under this approach will be:
Operating expense
$2,100,000
Depreciation
–1,785,000
EBT
$315,000
Taxes
126,000
Net income
$189,000
OCF
$1,974,000
The NPV under this method is:
NPV = –$6,190,000 + $1,974,000(PVIFA10%,4) + $160,000/1.104
And the IRR is:
0 = –$6,190,000 + $1,974,000(PVIFAIRR,4) + $160,000/(1 + IRR)4
So, this analysis still tells us the company should purchase the new machine. This is really the
same type of analysis we originally did. Consider this: Subtract the NPV of the decision to keep
the old machine from the NPV of the decision to purchase the new machine. You will get:
b. Even though the saved expenses are less than the cost of the machine, the cash flows are also
increased because of the higher depreciation of the new machine. The depreciation tax shield
23. We can find the NPV of a project using nominal cash flows or real cash flows. Either method will
result in the same NPV. For this problem, we will calculate the NPV using nominal cash flows. The
initial investment in either case is $1,910,000 since it will be spent today. We will begin with the
nominal cash flows. The revenues and production costs increase at different rates, so we must be
careful to increase each at the appropriate growth rate. The nominal cash flows for each year will be:
Year 0
Year 1
Year 2
Year 3
Revenues
$965,000.00
$1,013,250.00
$1,063,912.50
Costs
$425,000.00
442,000.00
459,680.00
Depreciation
272,857.14
272,857.14
272,857.14
EBT
$267,142.86
$298,392.86
$331,375.36
Taxes
90,828.57
101,453.57
112,667.62
Net income
$176,314.29
$196,939.29
$218,707.74
OCF
$449,171.43
$469,796.43
$491,564.88
Year 4
Year 5
Year 6
Year 7
Revenues
$1,117,108.13
$1,172,963.53
$1,231,611.71
$1,293,192.29
Costs
478,067.20
497,189.89
517,077.48
537,760.58
Depreciation
272,857.14
272,857.14
272,857.14
272,857.14
EBT
$366,183.78
$402,916.50
$441,677.08
$482,574.57
Taxes
124,502.49
136,991.61
150,170.21
164,075.35
Net income
$241,681.30
$265,924.89
$291,506.87
$318,499.21
OCF
$514,538.44
$538,782.03
$564,364.02
$591,356.36
Now that we have the nominal cash flows, we can find the NPV. We must use the nominal required
return with nominal cash flows. Using the Fisher equation to find the nominal required return, we get:
(1 + R) = (1 + r)(1 + h)
So, the NPV of the project using nominal cash flows is:
NPV = –$1,910,000 + $449,171.43/1.1235 + $469,796.43/1.12352 + $491,564.88/1.12353
Because the revenues and costs are growing annuities, we can find the present value of these
cash flows using the growing annuity equation. This will allow us to find the operating cash flow
using the tax shield approach. Since revenues and expenses are growing at different rates, we must
required return. 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:
PV of aftertax revenues = C{[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)]t}(1 – tC)
PV of aftertax revenues = $965,000{[1/(.07 – .05)] – [1/(.07 – .05)] × [(1 + .05)/(1 + .07)]7}(1 – .34)
PV of aftertax costs = C {[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)]t}(1 – tC)
PV of aftertax costs = $425,000{[1/(.07 – .04)] – [1/(.07 – .04)] × [(1 + .04)/(1 + .07)]7}(1 – .34)
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:
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 = 9,500
Year 2 quantity = 9,500(1 + .07) = 10,165
Year 3 quantity = 10,165(1 + .07) = 10,877
and operating cash flow each year will be:
Year 0
Year 1
Year 2
Year 3
Year 4
Year 5
Revenues
$636,500.00
$681,055.00
$728,728.85
$779,739.87
$834,321.66
Fixed costs
265,000.00
265,000.00
265,000.00
265,000.00
265,000.00
Variable costs
313,500.00
335,445.00
358,926.15
384,050.98
410,934.55
Depreciation
85,000.00
85,000.00
85,000.00
85,000.00
85,000.00
EBT
–$27,000.00
–$4,390.00
$19,802.70
$45,688.89
$73,387.11
Taxes
–9,180.00
–1,492.60
6,732.92
15,534.22
24,951.62
Net income
–$17,820.00
–$2,897.40
$13,069.78
$30,154.67
$48,435.49
OCF
$67,180.00
$82,102.60
$98,069.78
$115,154.67
$133,435.49
Equipment
–$425,000
NWC
–60,000
$60,000
Total CF
–$485,000
$67,180.00
$82,102.60
$98,069.78
$115,154.67
$193,435.49
So, the NPV of the project is:
NPV = –$485,000 + $67,180/1.15 + $82,102.60/1.152 + $98,069.78/1.153
+ $115,154.67/1.154 + $193,435.49/1.155
NPV = –$138,007.09
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:
PV of revenues = C{[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)]t}(1 – tC)
PV of revenues = $636,500{[1/(.15 – .07)] – [1/(.15 – .07)] × [(1 + .07)/(1 + .15)]5}
PV of revenues = $2,408,228.85
PV of variable costs = C{[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)]t}(1 – tC)
PV of variable costs = $313,500{[1/(.15 – .07)] – [1/(.15 – .07)] × [(1 + .07)/(1 + .15)]5}
PV of variable costs = $1,186,142.57
PV of fixed costs = C({1 – [1/(1 + r)]t }/r )
PV of fixed costs = $265,000({1 – [1/(1 + .15)]5 }/.15)
PV of fixed costs = $888,321.10
PV of depreciation = $284,933.18
NPV = –$485,000 + ($2,408,228.85 – 1,186,142.57 – 888,321.10)(1 – .34) + ($284,933.18)(.34)
+ $60,000/1.155
NPV = –$138,007.09
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 – 500,000)(.38)
Taxes on salvage value = –$190,000
Market price
$500,000
Tax on sale
–190,000
Aftertax salvage value
$310,000
Now we need to calculate the operating cash flow each year. Using the bottom up approach to
calculating operating cash flow, we find:
Year 0
Year 1
Year 2
Year 3
Year 4
Revenues
$3,220,000
$4,567,500
$5,127,500
$3,395,000
Fixed costs
725,000
725,000
725,000
725,000
Variable costs
483,000
685,125
769,125
509,250
Depreciation
1,566,510
2,089,150
696,070
348,270
EBT
$445,490
$1,068,225
$2,937,305
$1,812,480
Taxes
169,286
405,926
1,116,176
688,742
Net income
$276,204
$662,300
$1,821,129
$1,123,738
OCF
$1,842,714
$2,751,450
$2,517,199
$1,472,008
Capital spending
–$4,700,000
310,000
Land
–2,100,000
2,350,000
NWC
–450,000
450,000
Total cash flow
–$7,250,000
$1,842,714
$2,751,450
$2,517,199
$4,582,008
NPV = –$7,250,000 + $1,842,714/1.13 + $2,751,450/1.132 + $2,517,199/1.133
+ $4,582,008/1.134
NPV = $1,090,285.04