Chapter 12: Projects Risk and Uncertainty
Sensitivity Analysis
12.1
(a) Project cash flows based on most-likely estimates:
0 1 2 3
Income Statement
Labor Savings $45,000 $45,000 $45,000
Depreciation 20,000 32,000 9,600
Taxable Income $25,000 $13,000 $35,400
Income Tax (40%) 10,000 5,200 14,160
(b) With the required investment of $110,000
0 1 2 3
Income Statement
Labor Savings $50,744 $50,744 $50,744
Depreciation 23,000 36,800 11,040
Taxable Income $27,744 $13,944 $39,704
Income Tax (40%) 11,098 5,578 15,882
12.2
(a) Project cash flows based on most-likely estimates: without working capital
0 1 2 3 4
Income Statement
Labor Savings $35,000 $35,000 $35,000 $35,000
Depreciation 21,600 34,560 20,736 6,221
Taxable Income $13,400 $440 $14,264 $28,779
Income Tax (40%) 5,360 176 5,706 11,512
(b) Project cash flows based on most-likely estimates: with working capital
0 1 2 3 4
Income Statement
Labor Savings $35,000 $35,000 $35,000 $35,000
Depreciation 21,600 34,560 20,736 6,221
Taxable Income $13,400 $440 $14,264 $28,779
Income Tax (40%) 5,360 176 5,706 11,512
The project is still acceptable.
(c) Required annual savings (X): $43,370 through the table below.
0
1
2
3
4
Income Statement
Labor Savings
$43,370
$43,370
$43,370
$43,370
Depreciation
21,600
34,560
20,736
6,221
12.3
• Project’s IRR if the investment is made now:
PW( ) $500,000 $200,000( / , ,5) 0i P Ai=−+ =
12.4
(a) Economic building height
8,708
3,524
9,053
14,860
$13,062
$5,286
$13,580
$22,289
Cash Flow Statement
Net Income
13,062
5,286
13,580
22,289
Depreciation
21,600
34,560
20,736
6,221
30,000
34,662
39,846
34,316
56,463
n2 Floors 3 Floors 4 Floors 5 Floors
0($500,000) ($750,000) ($1,250,000) ($2,000,000)
Net Cash Flows
Best
i (%) 2 Floors 3 Floors 4 Floors 5 Floors Floor Plan
5$832,115 $687,721 $963,010 $1,987,770 5
6$787,037 $635,264 $873,011 $1,834,680 5
7$744,141 $585,441 $787,722 $1,689,448 5
8$703,298 $538,091 $706,879 $1,551,593 5
9$664,388 $493,067 $630,199 $1,420,666 5
10 $627,298 $450,230 $557,428 $1,296,250 5
Sensitivity Analysis
PW(i) as a Function of Interest Rate
(b) Effects of overestimation on resale value:
Resale
value
Present Worth as a Function of Number of Floors
2 Floors
3 Floors
4 Floors
5 Floors
Base
$465, 720
$264,644
$244, 495
$759,148
12.5
Note: In the problem statement, the current book value for the defender is given
as $13,000. This implies that the machine has been depreciated under the
(a)
• Cost of retaining the old machine
Keep the old machine
n4 5 6 7 8 9 10
Financial Data 0 1 2 3 4 5 6
Depreciation $2,000 $2,000 $2,000 $2,000 $2,000 $2,000
Book value $13,000 $11,000 $9,000 $7,000 $5,000 $3,000 $1,000
Market value $1,000
With the half–year convention mandated, the book value that should be used
in determining the gains tax for the defender (if sold now) is
Total depreciation
$1,000 $2,000 $2,000 $1,000= + + +
$6,000=
• Cost of replacing the old machine:
Buy a new machine
Financial Data n0 1 2 3 4 5 6
Depreciation $2,400 $3,840 $2,304 $1,382 $1,382 $691
Book value $12,000 9600 5760 3456 2074 691 0
Net Cash Flow ($2,800) $360 $936 $322 ($47) ($47) $876
PW (10%) = ($1,024) AE (10%) = ($235)
• Incremental cash flows:
Net Cash Flow Incremental Cash flow
(new-old)
n
New Machine
Old Machine
0
$2,800−
$2,800−
1
360
400−
760
2
4
5
(b) Sensitivity analysis: The answer remains unchanged. In fact, it (an increase in
(c) Break–even salvage value: Let X denote the minimum salvage value for the
old machine. Then, the net proceeds from sale of the old machine will be
Total depreciation
$6,000=
Book value
$14,000=
Salvage value
X=
12.6
NOTE to the first printing users: The bullet items in the problem statement
should be named (a) and (b). Please ignore parts (a) – (c) on Page 651.
(a) Net cash flows:
• Paying $0.60 per mile to a sales rep.
Provide a car to a sales rep.
Financial Data n0 1 2 3 4 5
Depreciation $5,000 $8,000 $4,800 $2,880 $1,440
PW (15%) = ($33,464) AEC (15%) = $9,983
It is cheaper to provide a car to a sales rep.
(b) Breakeven analysis:
12.7
Model A
Financial Data
n012345-7 8
Depreciation $857 $1,469 $1,049 $749 $536 $268
Book value $6,000 $5,143 $3,673 $2,624 $1,874 $1,339 $0
Net Cash Flow ($6,000) ($233) ($49) ($175) ($265) ($329) ($60)
PW (10%) = ($7,152) AE (10%) = ($1,341)
Model B
Financial Data
n012345-7 8 9 10
Depreciation $1,215 $2,082 $1,487 $1,062 $759 $379
Model A is preferred.
(b) Break–even annual O&M costs for machine A: Let X denotes a before–tax
annual operating cost for model.
PW(10%) $6,000 ($257 0.7 )( / ,10%,1)
X PF
=− +− +
Model A
Financial Data
n012345-7 8
Depreciation $857 $1,469 $1,049 $749 $536 $268
(c) With a shorter service life:
n
Net Cash Flow
Model A
Model B
0
-$6,000
-$8,500
1
-233
0
2
-49
3
-175
4
-265
5
2,883
-$5,216
-$6,464
12.8 Assuming that all old looms were fully depreciated
(a)
• Project cash flows: Alternative 1
Alternative 1
Financial Data
n0 1 2 3 4 5 6 7 8
Depreciation $306,669 $525,564 $375,342 $268,040 $191,641 $191,426 $191,641 $95,713
Book value $2,146,036 1,839,367 1,313,803 938,462 670,422 478,781 287,354 95,713 0
Cash Flow Statement
Investment ($2,108,836)
+(0.40)*Dn 122,667 210,226 150,137 107,216 76,656 76,571 76,656 38,285
+(0.60)*Sales 4,749,449 4,749,449 4,749,449 4,749,449 4,749,449 4,749,449 4,749,449 4,749,449
• Sensitivity analysis for alternative 1
Change
MARR
Labor cost
O&M
Revenue
-30%
$17,662,515
$14,663,394
$15,273,287
$8,661,943
-20%
$16,496,280
$14,599,529
$15,006,124
$10,598,562
-10%
$15,436,786
$14,535,665
$14,738,962
$12,535,181
$14,471,800
$14,471,800
$14,471,800
$14,471,800
$13,590,722
$14,407,936
$14,204,638
$16,408,419
$12,784,336
$14,344,071
$13,937,476
$18,345,038
$12,044,608
$14,280,207
$13,670,314
$20,281,657
• Project cash flows: Alternative 2
Alternative 2
Financial Data
n 0 1 2 3 4 5 6 7 8
Depreciation $160,083 $274,347 $195,930 $139,918 $100,038 $99,926 $100,038 $49,963
Book value $1,120,242 960,159 685,812 489,882 349,964 249,926 150,000 49,963 0
Market value 54,000
Gain/Loss 54,000
Net Cash Flow ($1,083,042) $3,347,835 $3,393,541 $3,362,175 $3,339,770 $3,323,817 $3,323,773 $3,323,817 $3,336,188
PW (18%) = $12,575,319 AE (18%) = $3,084,026
• Sensitivity analysis for alternative 2
Change
MARR
Labor cost
O&M
Revenue
-30%
$15,205,898
$12,885,109
$13,720,299
$7,103,571
-20%
$14,244,385
$12,781,846
$13,338,639
$8,927,487
-10%
$13,370,886
$12,678,582
$12,956,979
$10,751,403
$12,575,319
$12,575,319
$12,575,319
$12,575,319
$11,848,941
$12,472,055
$12,193,658
$14,399,234
$11,184,155
$12,368,792
$11,811,998
$16,223,150
$10,574,337
$12,265,528
$11,430,338
$18,047,066
(b) Sensitivity graph
$15,000,000
$20,000,000
Sensitivity Graph for Alt 2
$15,000,000
$20,000,000
Sensitivity Graph for Alt 1
12.9 Sensitivity graph
12.10
• The total cost in present value for 200 shift:
Electric
PW(10%) = $38,058
∵From the below table
0
1
2
3
4
5
6
7
Electric Power
O&M
($2,025)
($2,025)
($2,025)
($2,025)
($2,025)
($2,025)
($2,025)
Net cash flow
($2,025)
($2,025)
($2,025)
($2,025)
($2,025)
($2,025)
O&M
($8,100)
$1,400,000
$1,600,000
$1,800,000
$2,000,000
V = 6000
V = 5000
V = 4000
V = 3000
V = 2000
V = 1000
PW(10%) =
($69,345)
Gasoline
O&M
($7,372)
($7,372)
($7,372)
($7,372)
($7,372)
($7,372)
($7,372)
Initial cost
($20,107)
• PW(10%)s for 260 shift:
Electric
Diesel
PW(10%) = $40,285
• Sensitivity graph
$80,000
$90,000
Electric Power
LPG
Gasoline
Diesel Fuel
($20,107)
($7,372)
($7,372)
($7,372)
($5,372)
($54,971)
($5,928)
($5,928)
($5,928)
($5,928)
($22,263)
($22,263)
($5,928)
($5,928)
($3,728)
($49,994)
Break–Even Analysis
12.11
•PW of net investment:
0
$2,200,000 $600,000 $400,000 $3, 200,000P= − − − = −
•PW of after–tax revenue:
•PW of tax credit (shield) on depreciation:
n
Depreciation
Combined
Tax savings
Building
Furniture
1 $54,060 $57,160 $111,220(0.31) = $34,478
2 56,410 97,960 154,370(0.31) = 47,855
• PW of net proceeds from sale:
Property
(asset)
Cost basis
Salvage
value
Book value
Gains
(losses)
Gains
Taxes
Furniture
$400,000
$0
$0
$0
$0
X=
52.28%
12.12 Useful life of the old bulb:
14,600 /(19 365) 2.1 years× =
For computational simplicity, let’s assume a useful life of 2 years for the old
bulb. Then, the new bulb will last 4 years. Let X denote the price for the new
12.13
• PW of net investment:
0
$250,000P= −
• PW of tax credit (shield) on depreciation: (In this problem, we assume that the
purchasing cost of $250,000 does not include any land value. Therefore, the
entire purchasing cost will be the cost basis for depreciation purpose.)
Depreciation
Combined
n
Building
Tax savings
• PW of net proceeds from sale:
Total depreciation
$127,666=
$122,334=