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Sensitivity
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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