Sensitivity and Breakeven AnalysisLecture No. 29Professor C. S. ParkFundamentals of Engineering EconomicsCopyright 2005

Chapter 10Handling Project UncertaintyOrigin of Project RiskMethods of Describing Project RiskProbability Concepts for Investment DecisionsRisk-Adjusted Discount Rate Approach

In Engineering economics we predict cash flowsHow 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 RiskRisk: (in essence) the potential for lossProject Risk: variability in a projects NPWRisk Analysis: The assignment of probabilities to the various outcomes of an investment project

Methods of Describing Project RiskSensitivity 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 10.1Transmission-Housing Project by Boston Metal CompanyNew investment = $125,000Number of units = 2,000 unitsUnit Price = $50 per unitUnit variable cost = $15 per unitFixed cost = $10,000/YrProject Life = 5 yearsSalvage value = $40,000Income tax rate = 40%MARR = 15%

Example 10.1 - After-tax Cash Flow for BMCs Transmission-Housings Project Base Case

012345Revenues: Unit Price5050505050 Demand (units)2,0002,0002,0002,0002,000 Sales revenue$100,000$100,000$100,000$100,000$100,000Expenses: Unit variable cost$15$15$15$15$15 Variable cost30,00030,00030,00030,00030,000 Fixed cost10,00010,00010,00010,00010,000 Depreciation17,86330,61321,86315,6135,575Taxable Income$42,137$29,387$38,137$44,387$54,425Income taxes (40%)16,85511,75515,25517,75521,770Net Income$25,282$17,632$22,882$26,632$32,655

(Example 10.1, Continued)

Cash Flow Statement012345Operating activitiesNet income25,28217,63222,88226,63232,655Depreciation17,86330,61321,86315,6135,575Investment activitiesInvestment(125,000)Salvage40,000Gains tax(2,611)Net cash flow($125,500)$43,145$48,245$44,745$42,245$75,619

Sheet1

Example 10.1 BMC's Transmission-Housings Project

Input Data (Base):Output Analysis:

Unit Price ($)$50Output (NPW)$40,169

Demand2000

Var. cost ($/unit)$15

Fixed cost ($)$10,000

Salvage ($)$40,000

Tax rate (%)40%

MARR (%)15%

Income Statement

012345

Revenues:

Unit Price$50$50$50$50$50

Demand (units)20002000200020002000

Sales Revenue100,000100,000100,000100,000100,000

Expenses:

Unit Variable Cost$15$15$15$15$15

Variable Cost30,00030,00030,00030,00030,000

Fixed Cost10,00010,00010,00010,00010,000

Depreciation17,86330,61321,86315,6135,581

Taxable Income42,13729,38738,13744,38754,419

Income Taxes (40%)16,85511,75515,25517,75521,768

Net Income25,28217,63222,88226,63232,651

Cash Flow Statement

Operating Activities:

Net Income25,28217,63222,88226,63232,651

Depreciation17,86330,61321,86315,6135,581

Investment Activities:

Investment(125,000)

Salvage40,000

Gains Tax(2,613)

Net Cash Flow(125,000)43,14548,24544,74542,24575,619

Sheet2

Sheet3

Example 10.1 - Sensitivity Analysis for Five Key Input Variables Base

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$80,393Demand12,01019,04926,08833,13040,16947,20854,24761,28668,325Variable cost52,23649,21946,20243,18640,16937,15234,13531,11828,101Fixed cost44,19143,18542,17941,17540,16939,16338,15737,15136,145Salvage value37,78238,37838,97439,57340,16940,76541,36141,95742,553

Sensitivity graph BMCs transmission-housings project (Example 10.1)-20%-15%-10%-5%0%5%10%15%20%$100,00090,00080,00070,00060,00050,00040,00030,00020,00010,0000-10,000BaseUnit PriceDemandSalvage valueFixed costVariable 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) CostElectrical power:CR(10%) = ($30,000 - $3,000)(A/P, 10%, 7) + (0.10)$3,000 = $5,845LPG:CR(10%) = ($21,000- $2,000)(A/P, 10%, 7) + (0.10)$2,000 = $4,103Gasoline:CR(10%) = ($20,000-$2,000)(A/P, 10%, 7) + (0.10) $2,000 = $3,897Diesel 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 CostElectrical 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

Break-Even AnalysisExcel using a Goal Seek function

Analytical Approach

Excel Using a Goal Seek FunctionNPWBreakeven ValueDemand

Goal SeekFunctionParameters

Analytical Approach Unknown Sales Units (X)

012345Cash Inflows: Net salvage37,389 X(1-0.4)($50)30X30X30X30X30X 0.4 (dep)7,14512,2458,7456,2452,230Cash outflows: Investment-125,000 -X(1-0.4)($15)-9X-9X-9X-9X-9X -(0.6)($10,000)-6,000 -6,000-6,000-6,000-6,000Net Cash Flow-125,00021X + 1,14521X + 6,24521X +2,74521X +24521X +33,617

PW of cash inflowsPW(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 = 0Xb=1,430 units.

DemandPW of inflowPW of OutflowNPWX100.5650X- $44,49030.1694X + $145,11370.3956X-$100,6230$44,490$145,113100,62350094,773160,19865,4251000145,055175,28230,2271429188,197188,225281430188,298188,255431500195,338190,3674,9702000245,620205,45240,1682500295,903220,53775,366

OutflowBreak-Even Analysis Chart0 300 600 900 1200 1500 1800 2100 2400$350,000300,000250,000200,000150,000100,00050,0000-50,000-100,000ProfitLossBreak-even VolumeXb = 1430Annual Sales Units (X)PW (15%)Inflow

Scenario Analysis

Variable ConsideredWorst-CaseScenarioMost-Likely-CaseScenarioBest-CaseScenarioUnit demand1,6002,0002,400Unit price ($)485053Variable cost ($)171512Fixed Cost ($)11,00010,0008,000Salvage value ($)30,00040,00050,000PW (15%)-$5,856$40,169$104,295