Sensitivity and Breakeven Analysis

Lecture No. 29Professor C. S. ParkFundamentals of Engineering EconomicsCopyright © 2005

Chapter 10Handling Project Uncertainty Origin of Project Risk Methods of Describing

Project Risk Probability Concepts

for Investment Decisions

Risk-Adjusted Discount Rate Approach

In Engineering economics we predict cash flows

How do you know for sure that what you are claiming for interest rate, costs, revenues remain true???

Well for some situations you can be close enough to consider your “single point” analysis to be worthwhile.

For other you need to consider what is called RISK

We use the term risk to describe an investment project where cash flows are not known in advanced with certainty.

What to do: Instead of single point analysis, an array of outcomes and their probabilities or odds are to considered.

Origins of Project Risk

Risk: (in essence) the potential for loss

Project Risk: variability in a project’s NPW

Risk Analysis: The assignment of probabilities to the various outcomes of an investment project

Methods of Describing Project Risk Sensitivity Analysis: a means of identifying the

project variables which, when varied, have the greatest effect on project acceptability.

Break-Even Analysis: a means of identifying the value of a particular project variable that causes the project to exactly break even.

Scenario Analysis: a means of comparing a “base case” to one or more additional scenarios, such as best and worst case, to identify the extreme and most likely project outcomes.

Sensitivity Analysis – Example 10.1 Transmission-Housing Project by Boston Metal Company New investment = $125,000 Number of units = 2,000 units Unit Price = $50 per unit Unit variable cost = $15 per unit Fixed cost = $10,000/Yr Project Life = 5 years Salvage value = $40,000 Income tax rate = 40% MARR = 15%

Example 10.1 - After-tax Cash Flow for BMC’s Transmission-Housings Project – “Base Case”

0 1 2 3 4 5

Revenues:

Unit Price 50 50 50 50 50

Demand (units) 2,000 2,000 2,000 2,000 2,000

Sales revenue $100,000 $100,000 $100,000 $100,000 $100,000

Expenses:

Unit variable cost $15 $15 $15 $15 $15

Variable cost 30,000 30,000 30,000 30,000 30,000

Fixed cost 10,000 10,000 10,000 10,000 10,000

Depreciation 17,863 30,613 21,863 15,613 5,575

Taxable Income $42,137 $29,387 $38,137 $44,387 $54,425

Income taxes (40%) 16,855 11,755 15,255 17,755 21,770

Net Income $25,282 $17,632 $22,882 $26,632 $32,655

Cash Flow Statement 0 1 2 3 4 5

Operating activities

Net income 25,282 17,632 22,882 26,632 32,655

Depreciation 17,863 30,613 21,863 15,613 5,575

Investment activities

Investment (125,000)

Salvage 40,000

Gains tax (2,611)

Net cash flow ($125,500) $43,145 $48,245 $44,745 $42,245 $75,619

(Example 10.1, Continued)

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A B C D E F G

Example 10.1 BMC's Transmission-Housings Project

Income Statement0 1 2 3 4 5

Revenues: Unit Price 50$ 50$ 50$ 50$ 50$ Demand (units) 2000 2000 2000 2000 2000 Sales Revenue 100,000$ 100,000$ 100,000$ 100,000$ 100,000$ Expenses: Unit Variable Cost 15$ 15$ 15$ 15$ 15$ Variable Cost 30,000 30,000 30,000 30,000 30,000 Fixed Cost 10,000 10,000 10,000 10,000 10,000 Depreciation 17,863 30,613 21,863 15,613 5,581

Taxable Income 42,137$ 29,387$ 38,137$ 44,387$ 54,419$ Income Taxes (40%) 16,855 11,755 15,255 17,755 21,768

Net Income 25,282$ 17,632$ 22,882$ 26,632$ 32,651$

Cash Flow StatementOperating Activities: Net Income 25,282 17,632 22,882 26,632 32,651 Depreciation 17,863 30,613 21,863 15,613 5,581 Investment Activities: Investment (125,000) Salvage 40,000 Gains Tax (2,613)

Net Cash Flow (125,000)$ 43,145$ 48,245$ 44,745$ 42,245$ 75,619$

Example 10.1 - Sensitivity Analysis for Five Key Input Variables

Deviation -20% -15% -10% -5% 0% 5% 10% 15% 20%

Unit price $57 $9,999 $20,055 $30,111 $40,169 $50,225 $60,281 $70,337 $80,393

Demand 12,010 19,049 26,088 33,130 40,169 47,208 54,247 61,286 68,325

Variable cost 52,236 49,219 46,202 43,186 40,169 37,152 34,135 31,118 28,101

Fixed cost 44,191 43,185 42,179 41,175 40,169 39,163 38,157 37,151 36,145

Salvage value

37,782 38,378 38,974 39,573 40,169 40,765 41,361 41,957 42,553

Base

Sensitivity graph – BMC’s transmission-housings project (Example 10.1)

-20% -15% -10% -5% 0% 5% 10% 15% 20%

$100,000

90,000

80,000

70,000

60,000

50,000

40,000

30,000

20,000

10,000

0

-10,000

Base

Unit Price

Demand

Salvage value

Fixed cost

Variable cost

Example 10.2 - Sensitivity Analysis for Mutually Exclusive Alternatives

Electrical

Power

LPG

Gasoline

Diesel

Fuel

Life expectancy 7 year 7 years 7 years 7 years

Initial cost $30,000 $21,000 $20,000 $25,000

Salvage value $3,000 $2,000 $2,000 $2,200

Maximum shifts per year 260 260 260 260

Fuel consumption/shift 32 kWh 12 gal 11 gal 7 gal

Fuel cost/unit $0.05/kWh $1.00/gal $1.20/gal $1.10/gal

Fuel cost/shift $1.60 $12 $13.20 $7.7

Annual maintenance cost:

Fixed cost $500 $1,000 $1,200 $1,500

Variable cost/shift $5 $6 $7 $9

Capital (Ownership) Cost

Electrical power:CR(10%) = ($30,000 - $3,000)(A/P, 10%, 7) + (0.10)$3,000 = $5,845

LPG:CR(10%) = ($21,000- $2,000)(A/P, 10%, 7) + (0.10)$2,000 = $4,103

Gasoline:CR(10%) = ($20,000-$2,000)(A/P, 10%, 7) + (0.10) $2,000 = $3,897

Diesel fuel: CR(10%) = ($25,000 -$2,200)(A/P, 10%, 7) +(0.10) $2,200 = $4,903

Annual O&M Cost

Electrical power:

$500 + (1.60 + 5)M = $500 + 6.6M LPG:

$1,000 + (12 + 6)M = $1,000 + 18M Gasoline:

$800 + (13.2 + 7)M = $800 + 20.20M Diesel fuel:

$1,500 + (7.7 + 9)M = $1,500 + 16.7M

Annual Equivalent Cost

Electrical power:

AE(10%) = 6,345 + 6.6M LPG:

AE(10%) = 5,103 + 18M Gasoline:

AE(10%) = 4,697 + 20.20M Diesel fuel:

AE(10%) = 6,403 + 16.7M

0

2000

4000

6000

8000

10000

12000

0

20 40 60 80

100

120

140

160

180

200

220

240

260

Number of Shifts (M)

An

nu

al E

qu

ival

ent C

ost

($)

Electrical

LPG

Gasoline

Disel Fuel

Break-Even Analysis

Excel using a Goal Seek function

Analytical Approach

Excel Using a Goal Seek Function

Goal Seek

Set cell:

To value:

By changing cell:

Ok Cancel

? X

$F$5

0

$B$6

NPW

Breakeven Value

Demand

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A B C D E F G

Example 10.3 Break-Even Analysis

Input Data (Base): Output Analysis:

Unit Price ($) 50$ Output (NPW) $0Demand 1429.39Var. cost ($/unit) 15$ Fixed cost ($) 10,000$ Salvage ($) 40,000$ Tax rate (%) 40%MARR (%) 15%

0 1 2 3 4 5Income Statement Revenues: Unit Price 50$ 50$ 50$ 50$ 50$ Demand (units) 1429.39 1429.39 1429.39 1429.39 1429.39 Sales Revenue 71,470$ 71,470$ 71,470$ 71,470$ 71,470$ Expenses: Unit Variable Cost 15$ 15$ 15$ 15$ 15$ Variable Cost 21,441 21,441 21,441 21,441 21,441 Fixed Cost 10,000 10,000 10,000 10,000 10,000 Depreciation 17,863 30,613 21,863 15,613 5,581

Taxable Income 22,166$ 9,416$ 18,166$ 24,416$ 34,448$ Income Taxes (40%) 8,866 3,766 7,266 9,766 13,779

Net Income 13,299$ 5,649$ 10,899$ 14,649$ 20,669$

Cash Flow StatementOperating Activities: Net Income 13,299 5,649 10,899 14,649 20,669 Depreciation 17,863 30,613 21,863 15,613 5,581 Investment Activities: Investment (125,000) Salvage 40,000 Gains Tax (2,613)

Net Cash Flow (125,000)$ 31,162$ 36,262$ 32,762$ 30,262$ 63,636$

Goal SeekFunctionParameters

Analytical Approach Unknown Sales Units (X)

0 1 2 3 4 5

Cash Inflows:

Net salvage 37,389 X(1-0.4)($50) 30X 30X 30X 30X 30X 0.4 (dep) 7,145 12,245 8,745 6,245 2,230

Cash outflows:

Investment -125,000 -X(1-0.4)($15) -9X -9X -9X -9X -9X -(0.6)($10,000) -6,000 -6,000 -6,000 -6,000 -6,000

Net Cash Flow -125,000 21X +

1,145

21X + 6,245

21X +

2,745

21X +

245

21X +

33,617

PW of cash inflows

PW(15%)Inflow= (PW of after-tax net revenue)

+ (PW of net salvage value)

+ (PW of tax savings from depreciation

= 30X(P/A, 15%, 5) + $37,389(P/F, 15%, 5) + $7,145(P/F, 15%,1) + $12,245(P/F, 15%, 2)

+ $8,745(P/F, 15%, 3) + $6,245(P/F, 15%, 4)

+ $2,230(P/F, 15%,5)

= 30X(P/A, 15%, 5) + $44,490

= 100.5650X + $44,490

PW of cash outflows:

PW(15%)Outflow = (PW of capital expenditure_ + (PW) of after-tax expenses= $125,000 + (9X+$6,000)(P/A, 15%, 5)= 30.1694X + $145,113

The NPW:PW (15%) = 100.5650X + $44,490

- (30.1694X + $145,113)=70.3956X - $100,623.

Breakeven volume:

PW (15%) = 70.3956X - $100,623 = 0

Xb =1,430 units.

Demand

PW of

inflow

PW of

Outflow NPW

X 100.5650X

- $44,490

30.1694X

+ $145,113

70.3956X

-$100,623

0 $44,490 $145,113 100,623

500 94,773 160,198 65,425

1000 145,055 175,282 30,227

1429 188,197 188,225 28

1430 188,298 188,255 43

1500 195,338 190,367 4,970

2000 245,620 205,452 40,168

2500 295,903 220,537 75,366

Outflow

Break-Even Analysis Chart

0 300 600 900 1200 1500 1800 2100 2400

$350,000

300,000

250,000

200,000

150,000

100,000

50,000

0

-50,000

-100,000

Profit

Loss

Break-even Volume

Xb =

1430

Annual Sales Units (X)

PW

(1

5%)

Inflow

Scenario Analysis

Variable

Considered

Worst-

Case

Scenario

Most-Likely-Case

Scenario

Best-Case

Scenario

Unit demand 1,600 2,000 2,400

Unit price ($) 48 50 53

Variable cost ($) 17 15 12

Fixed Cost ($) 11,000 10,000 8,000

Salvage value ($) 30,000 40,000 50,000

PW (15%) -$5,856 $40,169 $104,295