Chapter 12.pdf

Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

Chapter 12 Projects Risk and Uncertainty Sensitivity Analysis 12.1 (a) Project cash flows based on most-likely estimates: 0 Income Statement Labor Savings Depreciation Taxable Income Income Tax (40%) Net Income Cash Flow Statement Cash From Operation: Net Income Depreciation Investmen ment&Salvage Salvage Gains Tax Net Cash Flow PW (15 (15%) =

(b) If MARR = 25%

1

2

3

$45,000 20,000 $25,000 10,000 $15,000

$45,000 32,000 $13,000 5,200 $7,800

$45,000 9,600 $35,400 14,160 $2 $21,240

15,000 20,000

7,800 32,000

21,240 9,600

-100,000 30,000 3,360 -100,000 $2, $2,741.84

35,000

39,800

64,200


12.2 (a) Project cash flows based on most-likely estimates: without working capital 0

1

2

3

4

Income Statement La Labor Savings Depreciation

$35,000 21,600

$35,000 34,560

$35,000 20,736

$35,000 6,221

Taxable Income Income Tax (40%)

$13,400 5,360

$440 176

$14,264 5,706

$28,779 11,512

$8,040

$264

$8,558

$17,268

8,040 21,600

264 34,560

8,558 20,736

17,268 6,221 30,000 -2,047

29,640

34,824

29,294

51,442

Net Income Cash Flow Statement Cash From Operation: Net Income Depreciation Investment&Salvage Gains Tax

-108,000

Net Cash Flow

-108,000

PW (10%) =

$4,870

The project is acceptable. (b) Project cash flows based on most-likely estimates: with working capital 0 Income Statement

1

2

3

4


12.2 (a) Project cash flows based on most-likely estimates: without working capital 0

1

2

3

4

Income Statement La Labor Savings Depreciation

$35,000 21,600

$35,000 34,560

$35,000 20,736

$35,000 6,221

Taxable Income Income Tax (40%)

$13,400 5,360

$440 176

$14,264 5,706

$28,779 11,512

$8,040

$264

$8,558

$17,268

8,040 21,600

264 34,560

8,558 20,736

17,268 6,221 30,000 -2,047

29,640

34,824

29,294

51,442

Net Income Cash Flow Statement Cash From Operation: Net Income Depreciation Investment&Salvage Gains Tax

-108,000

Net Cash Flow

-108,000

PW (10%) =

$4,870

The project is acceptable. (b) Project cash flows based on most-likely estimates: with working capital 0 Income Statement

1

2

3

4


(c) Required annual savings (X): $43,370 through the table below. 1

2

3

4

Income Statement Labor Savings Depreciation

0

$43,370 21,600

$43,370 34,560

$43,370 20,736

$43,370 6,221

Taxable Taxable Income Income Tax (40%)

$21,770 8,708

$8,810 3,524

$22,634 9,053

$37,149 14,860

Net Income Cash Flow Statement Cash From Operation: Net Income Depreciation Investment &Salvage Gains Tax

$13,062

$5,286

$13,580

$22,289

13,062 21,600

5,286 34,560

13,580 20,736

22,289 6,221 30,000 (2,047)

34,662

39,846

34,316

56,463

Net Cash Flow PW(18%) =

(108,000) (108,000) $0

12.3 • Project’s IRR if the investment is made now: PW (i ) = −$500, 000 + $200, 000(P / A ,i , 5) = 0

∴ i = 28.65%

• Let

X denote the new after-tax annual cash flow:

Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Net Cash Cash Flow s

n 0 1 2 3 4 5

2 Floors ($5 00 00 ,0 ,0 00 00) $ 19 9,10 0 $ 19 9,10 0 $ 19 9,10 0 $ 19 9,10 0 $ 79 9,10 0

3 Floors ($ 75 75 0, 0,0 00 00) $ 1 6 9 ,2 0 0 $ 1 6 9 ,2 0 0 $ 1 6 9 ,2 0 0 $ 1 6 9 ,2 0 0 $ 1 ,0 6 9 , 2 0 0

4 Floors ($1 ,2 ,2 50 50,0 00 00) $ 1 4 9 ,2 0 0 $ 1 4 9 ,2 0 0 $ 1 4 9 ,2 0 0 $ 1 4 9 ,2 0 0 $ 2 ,1 4 9 , 2 0 0

5 Floors ($ 2, 2,000 ,0 ,0 00 00 ) $ 37 8,15 0 $ 37 8,15 0 $ 37 8,15 0 $ 37 8,15 0 $ 3,3 78 ,1 50

Sensitivity Sensitivity Ana Analysis lysis PW(i) as a Function of Interest Rate

i (%) 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

2 Floors $832,115 $787,037 $744,141 $703,298 $664,388 $627,298 $591,924 $558,167 $525,937 $495,148 $465,720 $437,580 $410,657 $384,885 $360,205 $336,557 $313,889 $292,150 $271 292

3 Floors $687,721 $635,264 $585,441 $538,091 $493,067 $450,230 $409,452 $370,612 $333,599 $298,309 $264,644 $232,512 $201,829 $172,516 $144,496 $117,701 $92,066 $67,527 $44 029

4 Floors $963,010 $873,011 $787,722 $706,879 $630,199 $557,428 $488,330 $422,686 $360,291 $300,953 $244,495 $190,751 $139,565 $90,792 $44,298 ($46) ($42,357) ($82,746) ($121 319)

5 Floors $1,987,770 $1,834,680 $1,689,448 $1,551,593 $1,420,666 $1,296,250 $1,177,957 $1,065,427 $958,321 $856,326 $759,148 $666,513 $578,166 $493,867 $413,393 $336,533 $263,091 $192,883 $125 737

Best Floor Plan 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 2 2 2 2


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 alternative MACRS with half-year convention. In other words, the allowed depreciation is based on a 10-year recovery period with straight-line method (with $0 salvage). Note that, if you decide to retain the old machine, the current book value will be $13,000 as you continue to depreciate the asset without any adjustment. (a) Defender versus Challenger:

Keep the old machine n

Financial Data

Depreciation Book value Market value Gain/loss Removal cost Operating cost

4 0

5 6 7 8 9 10 1 2 3 4 5 6 $2,000 $2,000 $2,000 $2,000 $2,000 $2,000 $13,000 $11,000 $9,000 $7,000 $5,000 $3,000 $1,000 $1,000

2,000

2,000

2,000

2,000

1,500 2,000 2,000

800

($900) 800 800

Cash Flow Statement

Removal +(.4)*(Depreciation)

800

800

800


cost of retaining the old machine, or simply the new investment required to keep the old machine.

Buy a new machine Financial Data

n

Depreciation Book value Market value Gain/loss Operating cost

0

1 2 3 4 5 $2,400 $3,840 $2,304 $1,382 $1,382 $12,000 9600 5760 3456 2074 691

6 $691 0 2000 2000 1000

1000

1000

1000

1000

1000

960 -600

1,536 -600

922 -600

553 -600

553 -600

$360

$936

$322

($47)

($47) $876

Cash Flow Statement

Sale of old equipment Investment +(.4)*(Depreciation) -(1-0.40)*(Operating cost) Net proceeds from sale

9,200 -12,000

Net Cash Flow

($2,800)

PW (10%) = ($1,024)

AE (10%) = ($235)

• Incremental cash flows:

276 -600 1,200


(b) Sensitivity analysis: The answer remains unchanged. In fact, it (an increase in O&M) will make the challenger more attractive. IRR new−old = 24.25%, and PW (10%)new−old = $1, 280 (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 Taxable gain = X − $14,000 Net proceeds = X − (0.40)( X − $14, 000) = 0.6 X + $5, 600 To find the break-even salvage value, PW (10%)old = −$400( P / A,10%,5) − $300(P / F ,10%, 6) = −$1, 686 PW (10%)new

Let PW (10%)old

∴ X = $4,897

= −[$12, 000 − (0.6 X + $5, 600)] +$360( P / F ,10%,1) +  + $877(P / F ,10%, 6) = 0.6 X − $4, 624

X . = PW (10%) new and solve for


12.6 (a) Transmission distance of 5 miles: • Option 1 (Copper wire): 5 miles = 5 × 5,280 = 26,400 feet First cost = (1.692 + 0.013 × 2,000) × 26,400 = $731,069 Annual operating cost = $731,069(0.184) = $134,517

∴ PW (15%)1

= −$731, 069 − $134, 517(P / A,15%, 30) = −$1,614,305

• Option 2 (Fiber optics): Cost of ribbon = $15,000/mile × 5 miles = $75,000 Cost of terminators = $30,000 × 3 × 2 = $180,000 Cost of modulating system = ($12, 092 + $21, 217)(21)(2) = $1,398,978 Cost of repeater = $15,000 Total first cost = $75,000 + $180,000 + $1,398,978 + $15,000 = $1,668,978 Annual operating costs = $1,398,978(0.125) +$75,000(0.178) = $188,222

∴ PW (15%)2

= −$1, 668, 978 − $188, 222(P / A,15%, 30) = −$2,904,840

Option 1 is the better choice. (b) Either 10 miles or 25 miles of transmission distance:


12.7 (a) With infinite planning horizon: We assume that both machines will be available in the future with the same cost. Model A Financial Data n

Depreciation B ook value Market value Gain/Loss Operation Cost

0 $ 6,0 00

1 $857 $ 5,1 43

2 $1,469 $ 3,67 3

3 $1,049 $2 ,6 24

4 $749 $ 1,8 74

5-7 $536 $ 1,3 39

8 $268 $0 $500 $500 $700

$700

$700

$700

$700

$700

$257 ($490)

$441 ($490)

$315 ($490)

$225 ($490)

$161 ($490)

$ 80 ($490) $350

($233)

($49)

($175)

($265)

($329)

($60)

Cash Flow Statement

Investment +(.30)*(Depreciation) -(1-0.30)*(Operation cost) Net proceeds from sale

($6,000)

Net Cash Flow

($6,000) P W (1 0% ) = ($ 7, 15 2)

A E (1 0% ) = ($ 1, 34 1)

Model B Financial Data n

Depreciation Book value Market value Gain/Loss Operation Cost

0 $8,500

1 $1,215 $7,285

2 $2,082 $5,204

3 $1,487 $3,717

4 $1,062 $2,655

5-7 $759 $1,896

8 $379 ($0)

9

$520

$520

$520

$520

$364 ($364)

$624 ($364)

$446 ($364)

10

$520

$520

$520

$318 ($364)

$228 ($364)

$114 ($364)

$0 ($364)

$0 ($364) $700

($46)

($136)

($250)

($364)

$ 336

($0)

($0) $1,000 $1,000 $520

Cash Flow Statement

Investment +(.30)*(Depreciation) -(1-0.30)*(Operation cost) Net proceeds from sale

($8,500)

Net Cash Flow

($8,500) P W (1 0% ) = ($ 8, 62 7)

$0

$260

$82

A E (1 0% ) = ($ 1, 40 4)

Model A is preferred

Page | 9


(b) Break-even annual O&M costs for machine A: Let X denotes a before-tax annual operating cost for model.

= −$6, 000 + ($257 − 0.7 X )(P / F ,10%,1) + +($430 − 0.7 X )( P / F ,10%,8) = −$4,538 − 3.734 X AE (10%) A = −$851 − 0.7 X

PW (10%) A

Let AE (10%) A

= AE (10%) B , and solve for X .

−$851 − 0.7 X = −$1, 404 ∴ X = $791 per year Model A Financial Data n

Depreciation Book value Market value Gain/Loss Operation Cost

0 $6,000

1 $857 $5,143

2 $1,469 $3,673

3 $1,049 $2,624

4 $749 $1,874

5-7 $536 $1,339

$791

$791

$791

$791

$791

8 $268 $0 $500 $500 $791

$257 ($553)

$441 ($553)

$315 ($553)

$225 ($553)

$161 ($553)

$80 ($553)

Cash Flow Statement

Investment +(.30)*(Depreciation) -(1-0.30)*(Operation cost)

($6,000)


12.8 Assuming that all old looms were fully depreciated (a) • Project cash flows: Alternative 1

Alternative 1 Financial Data n

Depreciation Book value Market value Gain/Loss Annual sales Annual labor cost Annual O&M cost

0

1 $306,669 $2,146,036 1,839,367

2 $525,564 1,313,803

3 $375,342 938,462

4 $268,040 670,422

5 $191,641 478,781

5 $191,426 287,354

7 $191,641 95,713

7,915,748 261,040 1,092,000

7,915,748 261,040 1,092,000

7,915,748 261,040 1,092,000

7,915,748 261,040 1,092,000

7,915,748 261,040 1,092,000

7,915,748 261,040 1,092,000

7,915,748 261,040 1,092,000

122,667 4,749,449 -156,624 -655,200

210,226 4,749,449 -156,624 -655,200

150,137 4,749,449 -156,624 -655,200

107,216 4,749,449 -156,624 -655,200

76,656 4,749,449 -156,624 -655,200

76,571 4,749,449 -156,624 -655,200

76,656 4,749,449 -156,624 -655,200

8 $95,713 0 169,000 169,000 7,915,748 261,040 1,092,000

Cash Flow Statement

Investment +(0.40)*Dn +(0.60)*Sales -(0.60)*Labor -(0.60)*O&M Net proceeds from sale

($2,108,836) 38,285 4,749,449 -156,624 -655,200 101,400


• Project cash flows: Alternative 2

Alternative 2 Financial Data

n Depreciation Book value Market value Gain/Loss Annual sales Annual labor cost Annual O&M cost

0

1 $160,083 $1,120,242 960,159

2 $274,347 685,812

3 $195,930 489,882

4 $139,918 349,964

5 $100,038 249,926

5 $99,926 150,000

7 $100,038 49,963

7,455,084 422,080 1,560,000

7,455,084 422,080 1,560,000

7,455,084 422,080 1,560,000

7,455,084 422,080 1,560,000

7,455,084 422,080 1,560,000

7,455,084 422,080 1,560,000

7,455,084 422,080 1,560,000

8 $49,963 0 54,000 54,000 7,455,084 422,080 1,560,000

Cash Flow Statement

Investment +(0.40)*Dn +(0.60)*Sales -(0.60)*Labor -(0.60)*O&M Net proceeds from sale

($1,083,042) 64,033 4,473,050 -253,248 -936,000

109,739 4,473,050 -253,248 -936,000

78,372 4,473,050 -253,248 -936,000

55,967 4,473,050 -253,248 -936,000

40,015 4,473,050 -253,248 -936,000

39,970 4,473,050 -253,248 -936,000

40,015 4,473,050 -253,248 -936,000

19,985 4,473,050 -253,248 -936,000 32,400

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


(b) Sensitivity graph

Sensiti vity Graph for Alt 1

MARR Labor O&M Revenue

Sensitivity Graph for Alt 2 $20,000,000


12.9 Sensitivity graph

V = 6000

V = 5000

V = 4000 V = 3000 V = 2000 V = 1000

12.10 • NPW(10%)s for 200 shift:

= $38, 058 NPW (10%) LPG = $69,345 NPW (10%) = $54 971 NPW (10%) Electric


Gasoline O&M Initial cost

($7,372) ($7,372) ($7,372) ($7,372) ($7,372) ($7,372) ($7,372) ($20,107)

Salvage

$2,000

Net cash flow ($20,107) ($7,372) ($7,372) ($7,372) ($7,372) ($7,372) ($7,372) ($5,372) NPW(10%) = ($54,971)

Diesel Fuel O&M Initial cost

($5,928) ($5,928) ($5,928) ($5,928) ($5,928) ($5,928) ($5,928) ($22,263)

Salvage

$2,200

Net cash flow ($22,263) ($5,928) ($5,928) ($5,928) ($5,928) ($5,928) ($5,928) ($3,728) NPW(10%) = ($49,994)

• NPW(10%)s for 260 shift: = $40, 285 NPW (10%) LPG = $82, 635 NPW (10%)Gasoline = $64, 277 NPW (10%) Diesel = $57,192 NPW (10%) Electric

• Sensitivity graph Electric Power


Break-Even Analysis 12.11 • PW of net investment: P0

= −$2, 200, 000 − $600, 000 − $400, 000 = − $3, 200, 000

• PW of after-tax revenue: P1

= −$4, 000(365) X (1 − 0.31)( P / A,10%, 25) = $9,144,210 X

• PW of after-tax operating costs: P2

= −($230, 000 + $170, 000 X )(1 − 0.31)( P / A,10%, 25) = −$1, 440,526 − 1, 064, 737 X

• PW of tax credit (shield) on depreciation: n

1 2

Depreciation Building Furniture $54,060 $57,160 56,410

97,960

Combined Tax savings $111,220(0.31) = $34,478 154,370(0.31) = 47,855 126,370(0.31)


• PW of net proceeds from sale: Property (asset) Furniture Building Land

Cost basis

Salvage Book value value $0 $0

$400,000

Gains (losses) $0

Gains Taxes $0

2,200,000

0

794,450

(794,450)

(246,280)

600,000

2,031,813

600,000

1,431,813

443,862

Net proceeds from sale = $2,031,813 + $246,280 − $443,862 = $1,834,231 P4

= $1,834,231( P / F ,10%,25) = $169,292

PW (10%) = P0 + P1 + P2 + P3 + P4

= −$4,223,772 + 8,079,473 X =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


12.13 • PW of net investment: P0

= −$250,000

• PW of after-tax rental revenue: P1

= X (1 − 0.30)( P / A,15%, 20) = $4.3815 X

• PW of after-tax operation costs: P2

= −(1 − 0.30)$12, 000( P / A,15%, 20) = −$52,578

• 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 Building n 1 $6,143 2-19 6,410 20 6,143 P3

Combined Tax savings $6,143(0.30) = $1,843 6,410(0.30) = 1,923 6,143(0.30) = 1,843

= $1,843(P / F ,15%,1) + $1,923(P / A,15%,18)(P / F ,15%,1) +$1, 843( P / F 15%, 20)


• The break-even rental: PW (15%) = P0 + P1 + P2 + P3 + P4

= −$260,003 + 4.3815 X =0 X = $59,341

12.14 Let X denotes the additional annual revenue (above $14,000) for model A that is required to break even.

• Generalized cash flow for model A: End of Year Cash flow elements 0 1 2 3 4 5 6 Investment ($80,000) Net proceeds 12,000 +0.6 Rn +0.6X +0.6X +0.6X +0.6X +0.6X +0.6X +0.4 Dn $6,400 $10,240 $6,144 $3,686 $3,686 $1,843 -(0.6)O&M ($13,200) ($13,200) ($13,200) ($13,200) ($13,200) ($13,200) -$6,800 -$2,960 -$7,056 -$9,514 -$9,514 $634 Net cash flow ($80,000) +0.6X +0.6X +0.6X +0.6X +0.6X +0.6X

PW (20%) A

= −$100, 005.29 + 2 X

• Generalized cash flow for model B:


12.15 Let X denote the number of copies to break-even.

• A/T annual revenue = (0.6)[$0.05 + ($0.25 − $0.05)] X = 0.15 X • A/T O&M cost = −(0.60)[$300,000(12) + $0.10 X ] = −$2,160,000 − 0.06 X Depreciation tax credit = (0.40)[$85,714( P / F ,13%,1) + 

+$26, 775( P / F ,13%, 8)]( A / P,13%,10) = $29,285 CR(13%) = −$600, 000( A / P,13%,10) + $60, 000( A / F ,13%,10) = −$107,316 AE (13%) = 0.15 X − $2,160, 000 − 0.06 X + $29, 285− $107,316 = 0.09 X − $2, 238, 031 = 0 X = 24,867,011copies per year or 24,867,011/240 =103,613 copies

per day

Probabilistic Analysis 12.16

= −$8, 000, 000 + $1, 300, 000(P / A ,12%, 3) = −$4,877,619 PW (12%)moderate = −$8 000 000 + $2 500 000(P / A 12% 4) PW (12%)light


12.17 (a) 1.

A1 and A2 are mutually independent E[ PW (10%)] = −500 +

=0+ Var[ PW (10%)] 2.

200

500

+ 1

(1 + 0.1) (1 + 0.1) 502

2

502

+

(1 + 0.1) (1 + 0.1) 2

4

= $95.04

= 3773.65

A1 and A2 are partially correlated with ρ12 = 0.3 200 500 E[ PW (10%)] = −500 + + = $95.04 1 2 1 + 0.1 1 + 0.1

(

Var[ PW (10%)] = 0 +

) (

502

(1 + 0.1)

2

+

⎡ 0.3(50)(50) ⎤ + 2⎢ = 4900.62 3 ⎥ + (1 0.1) ⎦ (1 + 0.1) ⎣ 502

(b)

= Var[ PW (10%)] σ [ PW (10%)] = 3773.65 = 61.43 X − μ 0 − 95.04 z = = σ [ PW (10%)]

61.43 = −1.547 P( z < −1.547) = 0.06093 or 6.093% σ

)

4


(c) The mean and variance of the PW for Project 2:

= $0(0.24) + $400(0.20) + $1, 600(0.36) +$2, 400(0.20) = $1,136 2 2 Var[ PW (10%)]2 = (0 − 1,136) (0.24) + (400 − 1,136) (0.20) +(1,600 − 1,136)2 (0.36) + (2,400 − 1,136)2 (0.20) = 815,104 E[ PW (10%)]2

(d) No clear project dominance exists. However, Project 1 may be preferred over project 2 if we consider the probability of losing money or the possibility of realizing the worst situation. Project 1: $4,144 ± 3(1,858.62) whereas Project 2: $1,136 ± 3(902.83). 12.19 (a) Expected value criterion

• Option 1: E[ R]1

• Option 2:

= $2, 450(0.25) + $2, 000(0.45) + $1, 675(0.30) −$150( F / P, 7.5%,1) = $1,854


EPPI = 0.25($2, 288.75) + 0.45($1,875) + 0.30($1,875) = $1,978.44 EMV = $1,875 EVPI

= EPPI − EMV = $103.44

Note: EPPI (Expected Profit of Perfect Information) is the maximum profit that can be expected when you have the perfect information as we know what to do in each situation. 12.20 Let X denote the annual revenue in constant dollars and Y be the general inflation. (a) NPW as functions of X and Y : Cash elements Investment

End of Period 0

1

2

-$9,000

Salvage value

4,000(1+Y) 2

Gains tax

−0.4 ⎡⎣ 4,000(1+Y)2 - 4,000 ⎦⎤

(0.4) Dn

1,200

800

(0.6) Rn

0.6X(1+Y)

0.6 X (1 + Y ) 2

2,000(-Y)

2,000(1+Y)

working capital

-2,000


or

− 2, 000Y (1+Y)−1 + 0.6 X ]( P / F ,10%,1) +[2,400 + 2, 400(1+Y) −2 + 0.6 X + 2, 000(1+Y)-1 ]( P / F ,10%, 2)

PW (10%) = −$11, 000 + [1, 200(1+Y)

-1

(b) and (c) Mean and variance calculation: Note that the market interest rate is a random variable as the general inflation rate becomes a random variable. There are nine joint events for X and Y . For example of event No.2, the joint event where X = 10,000 and Y = 0.05 , we calculate the market interest rate and then evaluate the PW function with this market interest rate. i = i '+ f + i ' f = 0.10 + 0.05 + (0.1)(0.05) = 15.5% PW (15.5%) = −$11,000 + [1,200 − 2,000(0.05) + 0.6(10,000)(1.05)](P / F ,15.5%,1)

+[2,400(1.05)2 + 2,400 + 0.6(10,000)(1.05)2 + 2,000(1.05)] ×( P / F ,15.5%,2) = $5,722 You repeat the process for the remaining joint events.


Event No. 1 2 3 4 5 6 7 8 9

X 10,000 10,000 10,000 20,000 20,000 20,000 30,000 30,000 30,000

Y 0.03 0.05 0.07 0.03 0.05 0.07 0.03 0.05 0.07

i

Event No. PW(i%) 1 $5,877 2 $5,722 3 $5,574 4 $16,290 5 $16,136 6 $15,988 7 $26,704 8 $26,549 9 $26,401

P(x) 0.3 0.3 0.3 0.4 0.4 0.4 0.3 0.3 0.3

P(y)

E[PW]=

$ $ $ $ $ $ $ $ $

$16,137

0.133 0.155 0.177 0.133 0.155 0.177 0.133 0.155 0.177

0.25 0.5 0.25 0.25 0.5 0.25 0.25 0.5 0.25 Var[PW] = sigma[PW] =

A0 ($11,000) ($11,000) ($11,000) ($11,000) ($11,000) ($11,000) ($11,000) ($11,000) ($11,000)

A1 $7,320 $7,400 $7,480 $13,500 $13,700 $13,900 $19,680 $20,000 $20,320

A2 $13,372 $13,761 $14,157 $19,737 $20,376 $21,027 $26,102 $26,991 $27,896

PW(i%)*P(x,y) (PW(i%)-E[PW])^2*P(x,y) P(x,y) 0.075 $441 7,895,197 0.15 $858 16,270,806 0.075 $418 8,368,000 0.1 $1,629 2,345 0.2 $3,227 1 0.1 $1,599 2,238 0.075 $2,003 8,373,604 0.15 $3,982 16,259,759 0.075 $1,980 7,900,640 65,072,590 1 $16,137 65,072,590 $ 8,067


(b)

= (2, 000 − 1,900)2 (0.20) + (3, 000 − 1,900)2 (0.60) +(3,500 − 1,900)2 (0.20) = 1,240,000 2 2 Var[ PW ]2 = (1, 000 − 1,850) (0.30) + (2,500 − 1,850) (0.40) +(4,500 − 1,850)2 (0.30) = 2,492,500 Var[ PW ]1

Project 1 is still preferred, because of higher E[ PW ] with lower Var[ PW ] .

12.22 (a) Mean and variance calculations:

= ($100, 000)(0.20) + ($50, 000)(0.40) + (0)(0.40) = $40,000 E[ PW ] B = ($40, 000)(0.30) + ($10, 000)(0.40) + (− $10, 000)(0.30) = $13,000 2 2 Var[ PW ] A = (100, 000 − 40, 000) (0.20) + (50, 000 − 40, 000) (0.40) +(0 − 40, 000)2 (0.40) = 1,400, 000,000 (40, 000 13, 000)2 (0 30) + (10, 000 13, 000)2 (0 40) Var[ PW ] E[ PW ] A


(b) Assuming that both contracts are statistically independent from each other, Joint event ( PW A

> PWB )

($100,000,$40,000) ($100,000,$10,000) ($100,000,-$10,000) ($50,000,$40,000) ($50,000,$10,000) ($50,000,-$10,000) ($0,-$10,000)

Joint Probability (0.20)(0.30) = 0.06 (0.20)(0.40) = 0.08 (0.20)(0.30) = 0.06 (0.40)(0.30) = 0.12 (0.40)(0.40) = 0.16 (0.40)(0.30) = 0.12 (0.40)(0.30) = 0.12 Σ = 0.72

We could look at this problem from a different angle. That is, project A is certain not to lose money, and the second best outcome for project A is better than the best outcome for project B. So it actually doesn’t seem like a hard call.

12.23 (a) • Machine A: CR(10%) A

= ($60, 000 − $22, 000)( A / P ,10%, 6) + (0.10)($22, 000) = $10,924 E[ AE (10%)] A = ($5, 000)(0.20) + ($8, 000)(0.30) + ($10,000)(0.30) + ($12,000)(0.20) + $10,924 = $19,725


(b) Prob[ AE (10%) A

>

AE (10%) B ] :

Joint event (O & M A , O & M B ) ( AE A

> AE B )

Joint Probability

($10,000, $8,000)

($20,924, $19,042)

(0.30)(0.10) = 0.03

($12,000, $8,000)

($22,924, $19,042)

(0.20)(0.10) = 0.02

($12,000, $10,000)

($22,924, $21,042)

(0.20)(0.30) = 0.06

Σ = 0.11 12.24 (a) Mean and variance calculation (Note: For a random variable Y, which can be expressed as a linear function of another random variable X (say, Y = aX , where a is a constant) the variance of Y can be calculated as a function of variance of X , Var[Y ] = a 2Var[ X ] .

= −$5, 000 + $4, 000(P / A,15%, 2) = $1,502.84 E[ PW ] B = −$10, 000 + $6, 000(P / F ,15%,1) + $8, 000(P / F ,15%, 2) = $1,266.54 2 2 2 2 2 V [ PW ] A = 1, 000 + ( P / F ,15%,1) 1, 000 + ( P / F ,15%, 2) 1,500 E[ PW ] A


Decision–Tree Analysis 12.25 • Joint & Marginal probabilities:

H L Marginal probabilities Actual

H 0.12 0.07 0.19

Survey M 0.12 0.35 0.47

L 0.06 0.28 0.34

• Conditional probabilities:

Actual

H L

H 0.632 0.368

Survey M 0.255 0.745

L 0.176 0.824


(c) Decision tree

0.3 High 4 introduce

0 0

-0.2

1 No survey

4

0.7 Low

Event 3

-2 2

0

0

0

0

-2

0 Do not 0 0

0 0.632 High 4

introduce

0 0

1.792

0.19

4

0.368 Low

S

-2

#

1

0.14048

0

0

-2

1.792 Do not 0 0

0 0.255 High 4

introduce

0 0

-0.47

0.47 Do survey

0.745 Low

I

-2 2

-0.2

4

0.34048

0

0

-2

0 Do not 0 0

0 0.176 High 4

introduce

0

4


12.26 (a) Let’s define the symbols: P: Party is taking place NP: No party is planned TP: Tipster says “P” TNP: Tipster says “NP” Then, P(TP, NP ) = P (NP )P (TP / NP ) = (0.4)(0.2) = 0.08

(b) • Optimal decision without sample information: EMV = (0.6)(100) + (0.4)(−50) = 40 points.

Raid the dormitories.

• Joint & Marginal probabilities:

P NP Marginal Probability

Actual

Tipster says TP TNP 0.24 0.36 0.08 0.32 0.32 0.68

Marginal Probability 0.6 0.4 1


* Decision Tree

40

40

0.6

100

R 0.4 NR

-6

-50 -10

0.6

Do not take tips 40 62.5 62.5

Take tips

39.93

0.32

R NR

-7.5

0.4

0

0.75

100

0.25

-50 -10

0.75

TP 0.25 0.68 TNP

29.41 29.41

100 0.529

R 0.471 -5.3

0

-50


12.27 (a) Demand High case Income statement

0 Revenue Depreciation taxable income income tax(40%) Net income Cash flow statement Net income Depreciation Investment Salvage Gain tax Net cash flow PW(15%)=

1

2

1,000,000 12,287 987,714 395,085 592,628

1,000,000 12,821 987,180 394,872 592,308

1,000,000 12,821 987,180 394,872 592,308

592,628 12,287

592,308 12,821

592,308 12,821

3 ...14 1,000,000 12,821 987,180 394,872 592,308

592,308 12,821

15 1,000,000 12,287 987,714 395,085 592,628

592,628 12,287

(500,000) 100,000 83,504 (500,000)

604,915

605,128

605,128

500,000 12,287 487,714 195,085 292,628

500,000 12,821 487,180 194,872 292,308

500,000 12,821 487,180 194,872 292,308

292,628 12,287

292,308 12,821

292,308 12,821

605,128

788,419

500,000 12,821 487,180 194,872 292,308

15 500,000 12,287 487,714 195,085 292,628

292,308 12,821

292,628 12,287

3,060,763

Demand Medium case Income statement

0 Revenue Depreciation taxable income income tax(40%) Net income Cash flow statement Net income Depreciation Investment Salvage Gain tax Net cash flow PW(15%)= Demand Low case Income statement

1

2

3 ...14

(500,000) 100,000 83,504 (500,000) 1,306,552

304,915

305,128

305,128

305,128

488,419


•

EVPI 0 = EPPI − EV 0 = $1,440,849.7 − $1,222,349.8 = $218,499.9 EPPI =

$3,060,763 (.3) + $1,306,552 (.4) + $0(.3) = $1,440,849.7

∴ Could pay up to $218,499.9 to know the true state of nature. (b) • Joint / marginal probabilities: Survey says

Actual

Marginal Probability

High

Medium

Low

High

0.21

0.075

0.015

0.3

Medium

0.08

0.24

0.08

0.4

Low

0.015

0.06

0.225

0.3

0.305

0.375

0.320

1

Marginal Probability

• Conditional probabilities: Survey says


= EVPI 0 − EVPI = $218,499.9– $205,060.58= $13,439.32 EVSI = EV − EV 0

EVSI

e

e

• Decision Tree 0.3 Demand High 3,060,763 0

3060763

0.4 Demand medium

Open

1,306,552 0

1222349.8

0

1306552

0.3 Demand low

No survey 1 0

(728,333

1222349.8

0

-728333

Do not open 0

0 0.68852459 Demand High 3,060,763 0

3060763

0.26229508 Demand medium

Open

1,306,552 0

2414293.09

0.305 Sample info High # 1234789.12

0

1306552

0.04918033 Demand low 1

0

(728,333

2414293.09

0

-728333

Do not open 0

0 0.2 Demand High 3,060,763 0

3060763


Short Case Studies ST 12.1 • The EMV of the lottery by assuming that the lotto price is $1: • One ticket each for every possible combination of 6 numbers from 1 to 44: 44!

C (44, 6) =

Prize First Second Third Fourth Fail

= 7, 059, 052 6!(44 − 6)! Probability Payoff 0.0000001416 $27,007,364 - $1 0.0000323 $899 - $1 0.00149 $51 - $1 0.02381 $1 - $1 0.974668 $0 - $1 Total:

EMV $3.81 $0.03 $0.07 $0 -$0.97 $2.94

It is worth trying because of the positive EMV. • The probability is increased by 1/ 7,059,052 when you purchase one more ticket. ST 12.2 (a) Project cash flows: (a) Project cash flows based on most-likely estimates: 0 1 Income Statement

Revenue:

2

3

4

5

6

7

8-19

20


(b) Let X denote the steam charge per pound. Then, annual steam charge = 1,061,962(0.001 X )(365) = 387,616X n

1 2 3 4 5 6 7 8-19 20

Revenue $387,616 X $387,616 X $387,616 X $387,616 X $387,616 X $387,616 X $387,616 X $387,616 X $387,616 X

AE (10%) = $387, 616 X

Expenses -$660,886 -$741,277 -$836,725 -$949,847 -$1,083,699 -$1,241,839 -$1,428,415 -$1,637,000 -$8,337,000

− [$660,886(P / F ,10%,1) + 

+$8, 337, 000(P / F ,10%, 20)]( A / P ,10%, 20) = $387, 616 X − $1, 357, 918 =0 X = $3.503 per lb or $3,503 per thousand lbs (c) Sensitivity graph is not provided.


ST 12.3 (a) Project cash flows based on most-likely estimates:

0

1

2

3

4

5

5

7

8

Income Statement

Revenue: Bill savings Mile Savings Expenses: Depreciation

$3,000,000 $3,000,000 $3,000,000 $3,000,000 $3,000,000 $3,000,000 $3,000,000 $3,000,000

Taxable Income Income Tax (38%) Net Income

1,2 50,0 00

1,2 50,0 00

1,2 50,0 00

1,250,000

1,250,000

1,250,000

2,0 00,0 00

3,2 00,0 00

1,9 20,0 00

1,1 52,0 00

1,1 52,0 00

576,0 00

2,2 50,0 00

1,050,000

2,3 30,0 00

3,098,000

3,0 98,0 00

855,0 00

399,0 00

885,4 00

1,1 77,2 40

1,177,2 40

$1,395,000

1,250,000

1,250,000

3,674,000

4,250,000

4,2 50,0 00

1,396,1 20

1,615,0 00

1,6 15,0 00

$651,000 $1,444,600 $1,920,760 $1,920,760 $2,277,880 $2,635,000 $2,635,000

Cash Flow Statement

Cash From Operation: 1,3 95,0 00 651,0 00 1,444,6 00 1,920,760 1,920,760 2,277,880 2,635,000 2,635,000 Net Income 2,000,000 3,200,000 1,920,000 1,152,000 1,152,000 576,000 0 0 Depreciation Investment&Salvage -10,000,000 Net Cash Flow

-10,0 00,0 00

PW (18%) = $3,204,044 (b) Sensitivity analysis:

3,3 95,0 00

3,8 51,0 00

3,3 64,6 00

3,0 72,7 60

3,0 72,7 60

2,8 53,8 80

2,6 35,0 00

2,6 35,000


ST 12.4 (a), (b), and (c). Random Variables

Low

Most

High

Likely Annual Market size (units)

5,000

8,000

10,000

Growth rate (annual)

3%

5%

8%

Unit price

$80,000

$84,000

$86,000

Unit variable cost

$56,000

$60,000

$65,000

Fixed cost (annual)

$5,000,000

$8,000,000

$9,000,000

$6,000,000

$7,000,000

$8,000,000

excluding depreciation Salvage value

• Depreciated on a 7-year MACRS Depreciation: %

n

14.29 24.49 17.49 12.49

0 1 2 3 4

Dn

Bn

$7,859,500 $13,469,500 $9,619,500 $6,869,500

$55,000,000 $47,140,500 $33,671,000 $24,051,500 $17,182,000


Income statement

0 Revenue Expense Depreciation Taxable income Income taxes(35%) Net income

1 400,000,000 285,000,000 7,859,500 107,140,500 37,499,175 69,641,325

2 3 412,000,000 4 24,360,000 293,400,000 3 02,052,000 13,469,500 9,619,500 105,130,500 1 12,688,500 36,795,675 39,440,975 68,334,825 73,247,525

4 437,090,800 310,963,560 6,869,500 119,257,740 41,740,209 77,517,531

5 450,203,524 320,142,467 4,911,500 125,149,557 43,802,345 81,347,212

6 463,709,630 329,596,741 4,906,000 129,206,889 45,222,411 83,984,478

7 477,620,919 339,334,643 4,911,500 133,374,776 46,681,171 86,693,604

4

5

6

7

8 491,949,546 349,364,682 2,453,000 140,131,864 49,046,152 91,085,712

Cash flow statement

0 1 Operating activities Net income 69,641,325 Depreciation 7,859,500 Investment activities Investment -$55,000,000 Salvage Gains tax Net cash flow -$55,000,000 77,500,825 PW(15%)=

2

68,334,825 13,469,500

3 73,247,525 9,619,500

77,517,531 6,869,500

81,347,212 4,911,500

83,984,478 4,906,000

8

86,693,604 4,911,500

91,085,712 2,453,000

91,605,104

6,000,000 2,100,000 97,438,712

81,804,325

82,867,025

84,387,031

86,258,712

88,890,478

$324,589,088

Page | 40

Chapter 12.pdf

Recommend Documents