Multiple Investment AlternativesSensitivity

Analysis

ENGM 661

Given two or more investment alternatives, be able to identify the mutually exclusive alternatives.

Given two or more mutually exclusive investment alternatives, be able to determine the best alternative by the present worth method the annual worth method the incremental rate-of-return

Given a problem description, be able the breakeven point between two or more investment alternatives.

Given a cash flow, be able to perform a sensitivity analysis on one or two parameters of the cash flow.

Learning Objectives for tonight:

NPW > 0 Good Investment

EUAW > 0 Good Investment

IRR > MARR Good Investment

Note: If NPW > 0 EUAW > 0IRR > MARR

Summary

NPWA > NPWB Choose AMust use same planning horizon

EUAWA > EUAWB Choose ASame Planning Horizon implicit in computation

IRRA > IRRB Choose AMust use Incremental Rate-of-Return IRRB-A < MARR Choose A

Multiple Investments

Suppose we have two projects, A & B A B

Initial cost $50,000 $80,000Annual maintenance 1,000 3,000Increased productivity 10,000 15,000Life 10 10Salvage 10,000 20,000

Example

A

NPW(10) = -50 + 9(P/A,10,10) + 10(P/F,10,10)

Present Worth A

50

99

10

0

1 2 3 10

. . .

B

Present Worth B

80

1212

20

0

1 2 3 10

. . .

NPW(10) = -80 + 12(P/A,10,10) + 20(P/F,10,10)

NPWA > NPWB

Choose A

Conclusion

Equivalent Worth

50

99

10

0

1 2 3 10

. . .A

EUAW(10) = -50(A/P,10,10) + 9 + 10(A/F,10,10)

Equivalent Worth

1212

20

0

1 2 3 10

. . .B

EUAW(10) = -80(A/P,10,10) + 12 + 20(A/F,10,10)

Conclusion

EUAWA > EUAWB

Choose A

Example: Suppose MARR is 10%. Suppose also that we can invest in T-bill @15% or we can invest in a 5 year automation plan.

Different Planning Horizons

100

115

NPW = 115(1.1)-1 - 100= $4,545

100

30

51 2 3 4

NPW = 30(P/A,10,5) - 100= $13,724

A B

B

But this ignores reinvestment of T-bills for full5-year period.

Problem

0

5

100

201,135

NPW = 201.135(P/F,10,5) - 100= $24,889 A

Projects must becompared using same

Planning Horizon

Conclusion

Example; NPW

4,000

3

3,5004,500

A

NPW = -4 + 3.5(P/A, 10,3) + 4.5(P/F,10,3)

= -4 + 3.5(2.4869) + 4.5(.7513)

= 8.085

= $8,085

Example: NPW

5,000

3

3,000

5,000

6

B

NPW = -5 + 3(P/A,10,6) + 5(P/F,10,6)

Example: NPW

5,000

3

3,000

5,000

6

B

NPW = -5 + 3(P/A,10,6) + 5(P/F,10,6)

= -5 + 3(4.3553) + 5(.5645)

Example: NPW

5,000

3

3,000

5,000

6

B

NPW = -5 + 3(P/A,10,6) + 5(P/F,10,6)

= -5 + 3(4.3553) + 5(.5645)

= 10.888

= $10,888

Least Common MultipleShortest LifeLongest LifeStandard Planning Horizon

Planning Horizons

Example; NPW

A

4,000

3

3,5004,500

4,000

6

4,500

NPW = -4 -4(P/F,10,3) + 3.5(P/A,10,6) + 4.5(P/F,10,3)

+ 4.5(P/F,10,6)

Example: NPW

5,000

3

3,000

5,000

6

B

NPW = -5 + 3(P/A,10,6) + 5(P/F,10,6)

NPWA > NPWB

Choose A

Conclusion

EUAW

4,000

3

3,5004,500

A

EUAW = -4(A/P,10,3) + 3.5 + 4.5(A/F,10,3)

= -4(.4021) + 3.5 + 4.5(.3021)

= 3.251

= $3,251

Note: NPW = 3,251(P/A,10,6) = 3,251(4.3553) = $14,159

EUAW

5,000

3

3,000

5,000

6

B

EUAW = -5(A/P,10,6) + 3 + 5(A/F,10,6)

= -5(.2296) + 3 + 5(.1296)

= 2.500

= $2,500

Note: NPW = 2,500(P/A,10,6) = $10,888

Equivalent Uniform Annual Worth method implicitly assumes that you are comparing alternatives on a least common multiple planning horizon

EUAW

Two alternatives for a recreational facility are being considered. Their cash flow profiles are as follows. Using a MARR of 10%, select the preferred alternative.

Class Problem

EOY CF(A) CF(B)0 -11000 -50001 5000 20002 4000 30003 3000 40004 20005 1000

Critical Thinking

1 2 3 4 5

11

54

32

1A

B

1 2 3

5

432

Use Net Present Worth and least common multiple of lives to compare alternatives A & B.

Critical Thinking

1 2 3 4 5

11

54

32

1A

B

1 2 3

5

432

Use Net Present Worth and least common multiple of lives to compare alternatives A & B.

NPWA = 288(P/A,10,15)= 288(7.6061)= $2,191

NPWB = 926(P/A,10,15)= 926(7.6061)= $7,043

Spreadsheet123456789

1011121314151617181920212223

C D E

MARR = 10.0%

EOY CF(A) CF(B)0 (11,000) (5,000)1 5,000 2,0002 4,000 3,0003 3,000 (1,000)4 2,000 2,0005 (10,000) 3,0006 5,000 (1,000)7 4,000 2,0008 3,000 3,0009 2,000 (1,000)10 (10,000) 2,00011 5,000 3,00012 4,000 (1,000)13 3,000 2,00014 2,000 3,00015 1,000 4,000

NPV = 2,191 7,043PMT = 288 926

=NPV(E1,D5:D19)+D4 =-PMT($E1,15,D20)

Suppose we have two investment alternatives

Incremental Analysis

A

100

110

1

IRRA = 10%

B

200

226

1

IRRB = 13%

Suppose we have two investment alternatives

Incremental Analysis

A B

100

110

200

226

1 1

IRRA = 10% IRRB = 13%

IRRB > IRRA Choose B

Correction

Investment alternative B costs $200. If we foregoB for $100 invested in A, we have an extra $100 which can be invested at MARR. If MARR = 20%,

Correction

Investment alternative B costs $200. If we foregoB for $100 invested in A, we have an extra $100 which can be invested at MARR. If MARR = 20%,

A

100

110

1

IRRA = 15%

+

100

120

1=

200

230

1

Correction

B

200

226

1

IRRB = 13%

IRRA > IRRB Choose A

200

230

1

A

IRRA = 15%

Suppose we have $100,000 to spend and we have two mutually exclusive investment alternatives both of which yield returns greater than MARR = 15%.

Example 2

A

50,000

60,000

1

IRRA = 20%

B

90,000

106,200

1

IRRB = 18%

Example 2

A

50,000

60,000

1

IRRA = 20%

B

90,000

106,200

1

IRRB = 18%

IRRA > IRRB Choose A

Example 2

A

50,000

60,000

1

NPWA = -50 + 60(1.15)-1

= $2,170

B

90,000

106,200

1

NPWB = -90 + 106.2(1.15)-1

= $2,350

NPWB > NPWA Choose B

Remember, we have $100,000 available in funds so we could spend an additional $50,000 above alternative A or an additional $10,000 above alternative B. If we assume we can make MARR or 15% return on our money, then

Example 2

Example 2

if we invest in A, we have an extra $50,000 which can be invested at MARR (15%).

A

50,000

60,000

1

i = 20%

50,000

57,500

1

i = 15%

+ =

100,000

117,500

1

ic = 17.5%

Example 2

If we invest in B, we have an extra $10,000 which can be invested at MARR (15%).

B

90,000

106,200

1

i = 18%

10,000

11,500

1

i = 15%

+ =

100,000

117,700

1

ic = 17.7%

Example 2

B

100,000

117,700

1

IRRB = 17.7%

IRRcB > IRRcA Choose B

100,000

117,500

1

A

IRRA = 17.5%

Incremental Analysis

Incremental AnalysisMARR = 15%

t Drill X Drill Y Drill Z Y-X X-Y Z-X Z-Y

0 -39,000 -26,000 -45,000 13,000 -13,000 -6,000 -19,000

1 -12,000 -15,000 -9,000 -3,000 3,000 3,000 6,000

2 -12,000 -15,000 -9,000 -3,000 3,000 3,000 6,000

3 -12,000 -15,000 -9,000 -3,000 3,000 3,000 6,000

4 -12,000 -15,000 -9,000 -3,000 3,000 3,000 6,000

5 -5,000 -11,000 1,000 -6,000 6,000 6,000 12,000

NPW = ($75,746) ($74,294) ($70,198) $1,452 ($1,452) $5,548 $4,096

IRR = #NUM! #NUM! #NUM! 11% 11% 46% 23%

Differing Planning Horizons

Incremental AnalysisOption O B C

Initial Cost 0 9,000 12,000Net Cash 0 0 0Salvage 0 500 1,000

Life 0 4 8

Differing Planning Horizons

Cash Flows MARR = 15%

Period B C B2 C-B20 -9,000 -12,000 -9,000 -3,0001 0 0 0 0

2 0 0 0 03 0 0 0 04 500 0 -8,500 8,5005 0 0 06 0 0 07 0 0 08 1,000 500 500

IRR = #NUM! #NUM! #NUM! 30%NPV = ($8,500) ($11,000) ($17,000) $6,000

EUAW = (2,125) (1,375) (2,125) 750

ENGM 661Engineering Economics

forManagers

Break Even &Sensitivity

Motivation

Suppose that by investing in a new information system, management believes they can reduce inventory costs. Your boss asks you to figure out if it should be done.

Motivation

Suppose that by investing in a new information system, management believes they can reduce inventory costs. After talking with software vendors and company accountants you arrive at the following cash flow diagram.

1 2 3 4 5

100,000

25,000

i = 15%

Motivation

Suppose that by investing in a new information system, management believes they can reduce inventory costs. After talking with software vendors and company accountants you arrive at the following cash flow diagram.

1 2 3 4 5

100,000

25,000

NPW = -100 + 25(P/A,15,5) = -16,196

i = 15%

Motivation

Suppose that by investing in a new information system, management believes they can reduce inventory costs. After talking with software vendors and company accountants you arrive at the following cash flow diagram.

1 2 3 4 5

100,000

25,000

NPW = -100 + 25(P/A,15,5) = -16,196

i = 15%

Motivation

Boss indicates $25,000 per year savings is too low & is based on current depressed market. Suggests that perhaps $40,000 is more appropriate based on a more aggressive market.

1 2 3 4 5

100,000

40,000

Motivation

Boss indicates $25,000 per year savings is too low & is based on current depressed market. Suggests that perhaps $40,000 is more appropriate based on a more aggressive market.

1 2 3 4 5

100,000

40,000

NPW = -100 + 40(P/A,15,5) = 34,086

Motivation

Boss indicates $25,000 per year savings is too low & is based on current depressed market. Suggests that perhaps $40,000 is more appropriate based on a more aggressive market.

1 2 3 4 5

100,000

40,000

NPW = -100 + 40(P/A,15,5) = 34,086

Motivation

Tell your boss, new numbers indicate a go. Boss indicates that perhaps he was a bit hasty. Sales have fallen a bit below marketing forecast, perhaps a 32,000 savings would be more appropriate

1 2 3 4 5

100,000

32,000

Motivation

Tell your boss, new numbers indicate a go. Boss indicates that perhaps he was a bit hasty. Sales have fallen a bit below marketing forecast, perhaps a 32,000 savings would be more appropriate

1 2 3 4 5

100,000

32,000

NPW = -100 + 32(P/A,15,5) = 7,269

Motivation

Tell your boss, new numbers indicate a go. Boss indicates that perhaps he was a bit hasty. Sales have fallen a bit below marketing forecast, perhaps a 32,000 savings would be more appropriate

1 2 3 4 5

100,000

32,000

NPW = -100 + 32(P/A,15,5) = 7,269

Motivation

Tell your boss, new numbers indicate a go. Boss leans back in his chair and says, you know . . . .

Motivation

Tell your boss, new numbers indicate a go. Boss leans back in his chair and says, you know . . . .

I’ll do anything, justtell me what numbersyou want to use!

Motivation

1 2 3 4 5

100,000

A

NPW = -100 + A(P/A,15,5) > 0

Motivation

1 2 3 4 5

100,000

A

NPW = -100 + A(P/A,15,5) > 0

A > 100/(A/P,15,5) > 29,830

A < 29,830

A > 29,830

Motivation

1 2 3 4 5

100,000

A

Break-Even Analysis

Site Fixed Cost/Yr Variable CostA=Austin $ 20,000 $ 50 S= Sioux Falls 60,000 40 D=Denver 80,000 30

TC = FC + VC * X

Break-Even (cont)

Break-Even Analysis

0

50,000

100,000

150,000

200,000

250,000

0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000

Volume

Tota

l C

ost

Austin

S. Falls

Denver

Class Problem A firm is considering a new product line and the following data have been recorded:

Sales price $ 15 / unitCost of Capital $300,000Overhead $ 50,000 / yr.Oper/maint. $ 50 / hr.Material Cost $ 5 / unitProduction 50 hrs / 1,000 unitsPlanning Horizon 5 yrs.MARR 15%

Compute the break even point.

Class Problem

Profit Margin = Sale Price - Material - Labor/Oper.

= $15 - 5 - $50 / hr

= $ 7.50 / unit

50 hrs1000 units

Class Problem

Profit Margin = Sale Price - Material - Labor/Oper.

= $15 - 5 - $25 / hr

= $ 7.50 / unit

50 hrs1000 units

1 2 3 4 5

300,000

7.5X

50,000

Class Problem

Profit Margin = Sale Price - Material - Labor/Oper.

= $15 - 5 - $25 / hr

= $ 7.50 / unit

50 hrs1000 units

1 2 3 4 5

300,000

7.5X

50,000

300,000(A/P,15,5) + 50,000 = 7.5X

139,495 = 7.5X

X = 18,600

Suppose we consider the following cash flow diagram:

NPW = -100 + 35(P/A,15,5) = $ 17,325

Sensitivity

1 2 3 4 5

100,000

35,000

i = 15%

Suppose we don’t know A=35,000 exactly but believe we can estimate it within some percentage error of + X.

Sensitivity

1 2 3 4 5

100,000

35,000(1+X) i = 15%

Then,

EUAW = -100(A/P,15,5) + 35(1+X) > 0

35(1+X) > 100(.2983)

X > -0.148

Sensitivity

1 2 3 4 5

100,000

35,000(1+X)

i = 15%

Sensitivity (cont.)

NPV vs. Errors in A

(20,000)

(10,000)

0

10,000

20,000

30,000

40,000

50,000

-0.30 -0.20 -0.10 0.00 0.10 0.20

Error X

NP

V

Now suppose we believe that the initial investment might be off by some amount X.

Sensitivity (Ao)

1 2 3 4 5

100,000(1+X)

35,000

i = 15%

Sensitivity (Ao)

NPV vs Initial Cost Errors

(20,000)

(10,000)

0

10,000

20,000

30,000

40,000

50,000

-0.30 -0.20 -0.10 0.00 0.10 0.20

Error X

NP

V

Sensitivity (A & Ao)

NPV vs Errors

(20,000)

(10,000)

0

10,000

20,000

30,000

40,000

50,000

-0.30 -0.20 -0.10 0.00 0.10 0.20

Error X

NP

V

Errors in initial cost

Errors in Annual receipts

Now suppose we believe that the planning horizon might be shorter or longer than we expected.

Sensitivity (PH)

1 2 3 4 5 6 7

100,000

35,000i = 15%

Sensitivity (PH)

NPV vs Planning Horizon

(30,000)

(20,000)

(10,000)

0

10,000

20,000

30,000

40,000

50,000

0 1 2 3 4 5 6 7

NPV

PH

Sensitivity (Ind. Changes)NPV vs Errors

(20,000)

(10,000)

0

10,000

20,000

30,000

40,000

50,000

-0.30 -0.20 -0.10 0.00 0.10 0.20

Error X

NP

V

Errors in initial cost

Errors in Annual receipts

n=3

n=7

Planning Horizon

MARR

Multivariable Sensitivity

Suppose our net revenue is composed of $50,000 in annual revenues which have an error of X and $20,000 in annual maint. costs which might have an error of Y (i=15%).

1 2 3 4 5

100,000

50,000(1+X)

20,000(1+Y)

Multivariable Sensitivity

Suppose our net revenue is compose of $50,000 in annual revenues which have an error of X and $20,000 in annual maint. costs which might have an error of Y (i=15%).

1 2 3 4 5

100,000

50,000(1+X)

20,000(1+Y)

Multivariable SensitivitySuppose our net revenue is compose of $50,000 in annualrevenues which have an error of X and $20,000 in annualmaint. costs which might have an error of Y.

1 2 3 4 5

100,000

50,000(1+X)

20,000(1+Y)

You Solve It!!!

You Solve It!!!

Multivariable Sensitivity

EUAW = -100(A/P,15,5) + 50(1+X) - 20(1+Y) > 0

50(1+X) - 20(1+Y) > 29.83

1 2 3 4 5

100,000

50,000(1+X)

20,000(1+Y)

Multivariable Sensitivity

EUAW = -100(A/P,15,5) + 50(1+X) - 20(1+Y) > 0

50(1+X) - 20(1+Y) > 29.83

50X - 20Y > -0.17

X > 0.4Y - 0.003

1 2 3 4 5

100,000

50,000(1+X)

20,000(1+Y)

Multivariable SensitivitySimultaneous Errors (Rev. vs. Cost)

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

Error X

Err

or Y

Unfavorable

Favorable

+ 10%

Mutually Exclusive Alt.

Suppose we work for an entity in which the MARR is not specifically stated and there is some uncertainty as to which value to use. Suppose also we have the following cash flows for 3 mutually exclusive alternatives.t A1t A2t A3t

0 (50,000) (75,000) (100,000)1 18,000 25,000 32,000 2 18,000 25,000 32,000 3 18,000 25,000 32,000 4 18,000 25,000 32,000 5 18,000 25,000 32,000

Mutually Exclusive Alt.t A1t A2t A3t

0 (50,000) (75,000) (100,000)1 18,000 25,000 32,000 2 18,000 25,000 32,000 3 18,000 25,000 32,000 4 18,000 25,000 32,000 5 18,000 25,000 32,000 MARR = NPV1 NPV2 NPV3

4.0% 30,133 36,296 42,458 6.0% 25,823 30,309 34,796 8.0% 21,869 24,818 27,767 10.0% 18,234 19,770 21,305 12.0% 14,886 15,119 15,353 14.0% 11,795 10,827 9,859 16.0% 8,937 6,857 4,777 18.0% 6,289 3,179 69 20.0% 3,831 (235) (4,300)

Mutually Exclusive Alt.

NPV vs. MARR

(10,000)

0

10,000

20,000

30,000

40,000

50,000

0.0% 5.0% 10.0% 15.0% 20.0%

MARR

NP

V

NPV1

NPV2

NPV3