1. Engineering Economic Analysis
Chapter 10
Uncertainty in Future Events
Donald G. Newnan
San Jose State University
Ted G. Eschenbach
University of Alaska Anchorage
Jerome P. Lavelle
North Carolina State University
Neal A. Lewis
University of New Haven
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2. Chapter Outline Estimates & their Use in Economic Analysis
Range of Estimates
Probability
Joint Probability Distributions
Expected Value
Economic Decision Trees
Risk
Risk versus Return
Simulation
Real Options
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3. Learning Objectives Use a range of estimates to evaluate a project
Describe possible outcomes with probability distributions
Combine probability distributions for individual variables into joint probability distributions.
Use expected values for economic decision making
Use economic decision trees
Measure & consider risk
Understand how simulation can be used to evaluate economic decisions
Understand how diversification reduces risk for investments & project portfolios
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4. Vignette: Video Game Development & Uncertainty
Requirements for gaming software:
Work with wide range of PC configurations & internet speeds
Constraints:
Time & budget
Uncertainties:
Critical issues found in test runs
New operating system patches
Simple to complex upgrades
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5. Vignette: Video Game Development & Uncertainty
How to quantify uncertainty in a software development effort?
How to estimate impact on project’s schedule & costs of uncertainty in software testing program?
How to prepare for today’s applications & future unforeseen applications?
How to economically justify projects whose payoff is improved recruiting & training rather than product sales?
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6. Estimates in Economic Analysis
Economic analysis requires evaluating future consequences
Usually, a single value selected as best estimate
Economic analysis assumes estimates
Breakeven analysis examines impact of variability of one estimate
What-if analysis evaluates scenarios with multiple uncertainties
This chapter uses probabilities to better address uncertainty
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7. Why Do Sensitivity Analysis?
Data is uncertain
Make better decisions
Decide which data merits refinement
Focus managerial attention during implementation
All of the above
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8. Why Do Sensitivity Analysis?
Data is uncertain
Make better decisions
Decide which data merits refinement
Focus managerial attention during implementation
All of the above
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9. Using a Range of Estimates in Economic Analysis
More realistic to describe variables with a range of possible values
A range can include
Optimistic
Most likely
Pessimistic
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10. Example 10-1 Using a Range of Estimates in Economic Analysis
Optimistic
Most Likely
Pessimistic
Cost
$950
$1000
$1150
Net annual benefit
$210
$200
$170
Useful life, in years 12 10 8
Salvage value
$100 $0
For most likely scenario
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11. Mean Value from a Range To calculate a mean or average from a range
Approximation using beta distribution
Widely used in project management
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12. Example 10-2 Using a Range of Estimates in Economic Analysis
Optimistic
Most Likely
Pessimistic
Mean Value
Cost
$950
$1000
$1150
$1016.7
Net annual benefit
$210
$200
$170
$196.7
Useful life, years 12 10 8
Salvage value
$100 $0
$16.7
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13. Probability
Likelihood of outcome or long-run relative frequency of outcome’s occurrence in many trials
Probabilities must follow:
Engineering econ. commonly uses discrete probability distributions with few outcomes
(10-2)
(10-3)
Probabilities can be based on data, expert judgment, or both.
All data in economic analysis may have some level of uncertainty.
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14. Example 10-3 Probability 𝑃 𝑂𝑢𝑡𝑐𝑜𝑚𝑒 𝑗 =1 𝑃 $10000 =1−0.6−0.3=0.1
Optimistic
Most Likely
Pessimistic
Annual benefit
$10,000
$8000
$5000
Probability
P($10,000)
60%
30%
Life, in years 9 6
P(9)
P(6)=2P(9)
𝑃 𝑂𝑢𝑡𝑐𝑜𝑚𝑒 𝑗 =1
𝑃 $10000 =1−0.6−0.3=0.1
𝑃 9 +𝑃 6 =1; & 𝑃 6 =2𝑃 9
𝑃 9 = ; 𝑃 6 = 2 3
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15. Joint Probability Distributions
Joint probability distribution describes likelihood of outcomes from 2 or more variables. Each random variable has own probability distribution.
If events A & B are independent, the joint probability for both A & B to occur is:
P(A & B) = P(A) 𝗑 P(B) (10-4)
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16. Example 10-4 Joint Probability Distribution
Annual Benefit
Probability
Life
$5,000
0.3 6
0.67
8,000
0.6
10,000
0.1
5,000 9
0.33
NPW
Joint
Probability
-$3,224
0.200
9,842
0.400
18,553
0.067
3,795
0.100
21,072
32,590
0.033
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17. Expected Value
Expected value = weighted average of outcomes times their probabilities
(10-5)
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18. Example 10-5 Expected Value
Optimistic
Most Likely
Pessimistic
Annual benefit
$10,000
$8000
$5000
Probability
10%
60%
30%
Life, in years 9 6
33.3%
66.7%
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19. EV(Revenue) EV(revenue) = Revenue/year $5900 $6000 $6270 $6333
High: $10,000 & P = .2
Most likely: $6000 & P = .5
Low: $3000 & P = .3
EV(revenue) =
$5900
$6000
$6270
$6333
I don’t know
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20. EV(Revenue) EV(revenue) = Revenue/year $5900 $6000 $6270 $6333
High: $10,000 & P = .2
Most likely: $6000 & P = .5
Low: $3000 & P = .3
EV(revenue) =
$5900
$6000
$6270
$6333
I don’t know
=0.2(10,000)+0.5(6000)+0.3(3000)
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21. Uncertainty in Real World
What is likely?
Initial cost more likely to be high than low
Sales more likely lower than target than higher
Product life likely shorter than longer
All of the above
None of the above
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22. Uncertainty in the real world
What is likely?
Initial cost more likely to be high than low
Sales more likely lower than target than higher
Product life likely shorter than longer
All of the above
None of the above
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23. Example 10-6 Joint Probability Distribution
Annual Benefit
Probability
Life
$5,000
0.3 6
0.67
8,000
0.6
10,000
0.1
5,000 9
0.33 PW
Joint
Probability
PW × Joint
-$3,224
0.200
-$645
9,842
0.400
3,937
18,553
0.067
1,237
3,795
0.100
380
21,072
4,214
32,590
0.033
1,086
EV(PW)=10,209
More accurate & descriptive of possibilities than Example 10-5
approximation using EV’s of $7300 & 7 years
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24. Example 10-7 Expected Value
Dam Height
First Cost
Annual
P(Flood>Height)
Damage if Flood Occurs
No dam $0
0.25
$800,000 20
700,000
0.05
500,000 30
800,000
0.01
300,000 40
900,000
0.002
200,000
Dam Height (ft)
EUAC of First Cost
Expected Annual Flood Damage
Total Expected EUAC
No dam $0
$200,000 20
38,344
25,000
63,344 30
43,821
3,000
46,821 40
49,299
400
49,699
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25. Economic Decision Trees
Economic decision trees display all decisions & all possible outcomes with their probabilities.
Decision Node D1 D2 DX
Outcome Node
Chance Node C1 C2 CY p1 p2 py
Pruned Branch
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26. Example 10-8 t=0 t=1 t=2, …, 7. Net Rev.=$0 Salvage =$550K Terminate
4. Net Revenue
Year 1=$100K
Low Volume
P=0.3
Continue
8. Net Revenue
$100K/year
2. Volume for
New Product
Med. Volume
P=0.6
5. Net Revenue
Year 1=$200K Year 2…n = $200K
Yes
First cost=$1M
High Volume
P=0.1
9. Net Revenue
=$600K/year
Expand
First cost=$800K
6. Net Revenue
Year 1=$400K
Build New
Product No
Continue
10. Net Revenue
=$400K/year
3. $0
t=0
t=1
t=2, …,
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27. Example 10-8, Partially solved
7. Revenue=$0
Terminate
PW1 = $550K
4. Net Revenue
Year 1=$100K
Low Volume
P=0.3
Continue //
8.Revenue=$100K/yr
PW1 = $486.6K
2. Volume for
New Product
Med. Volume
P=0.6
5. Revenue Year 1, 2..8 =$200K
Yes
First cost=$1M
High Volume
P=0.1
9. Revenue=$600K/yr
Expand
First cost=$800K
PW1 = $2120.8K
6. Net Revenue
Year 1=$400K
Build New
Product
Continue // No
10.Revenue=$400K/yr
PW1 = $1947.2K
3. $0
t=0
t=1
t=2, …,
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28. Example 10-8, Solved // // // t=0 t=1 t=2, …, PW = $590.9K
7. Revenue=$0
Terminate
PW1 = $550K
4. Net Revenue
Year 1=$100K
Low Volume
P=0.3
Continue //
8.Revenue=$100K/yr
PW1 = $486.6K
PW = $1067K
2. Volume for
New Product
Med. Volume
P=0.6
5. Revenue Year 1, 2..8 =$200K
EV = $1046.6K
Yes
First cost=$1M
High Volume
P=0.1
9. Revenue=$600K/yr
PW = $2291.7K
Expand
First cost=$800K
PW1 = $2120.8K
6. Net Revenue
Year 1=$400K
Build New
Product No
Continue //
10.Revenue=$400K/yr //
PW = $46.6K
PW1 = $1947.2K
3. $0
t=0
t=1
t=2, …,
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29. Example 10-9 No accident $0 P=0.9 EV=$36 Small accident $300 P=0.07
(<$500 deductible)
Buy Insurance
$800
Totaled
P=0.03
$500
No accident
P=0.9 $0
Self-Insure $0
EV=$411
Small accident
P=0.07
$300
Totaled
P=0.03
Decision node
Buy = = $836
Self-insure = $411
$13,000
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30. Risk Risk is chance of getting an outcome other than expected value
Measures of risk
Probability of a loss
Standard deviation (σ)
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31. Example 10-10 Risk Continued from Example 10-9
$800 is constant if buy insurance & arithmetic simpler if not included in σ calculations
Buying insurance EV ≈ 2x self insure EV but σ of self insure is 20X larger
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32. Example 10-11 Risk (Example 10-4 continued)
Annual Benefit
Prob.
Life (years)
$5,000
0.3 6
0.67
8,000
0.6
10,000
0.1
5,000 9
0.33 PW
Joint
Prob.
-$3,224
0.200
9,842
0.400
18,553
0.067
3,795
0.100
21,072
32,590
0.033
PW x Probability
-$645
3,937
1,237
380
4,214
1,086
PW2 x Probability
2,079,480
38,747,954
22,950,061
1,442,100
88,797,408
35,392,740
$10,209
=EV(PW)
$189,409,745
=EV(PW2) =
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33. Example 10-12 Risk vs. Returns
Project
IRR
Std. Dev. 1
13.10%
6.50% 2
12.00%
3.90% 3
7.50%
1.50% 4
3.50% 5
9.40%
8.00% 6
16.30%
10.00% 7
15.10%
7.00% 8
15.30% F
4.00%
0.00%
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34. Simulation Simulation is advanced approach to consider risk
Economic simulation uses random sampling from probability distributions
For each iteration, all variables with a probability distribution are randomly sampled
Results of all iterations combined to create a probability distribution for outcome
Perform simulation
By hand with table of random number
Using Excel functions
Simulation programs such or Crystal Ball
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35. Spreadsheet & Economic Simulation
Excel Functions
Purpose
RAND ( )
Returns an evenly distributed random number that is
0 ≤ x < 1
New random number returned every time worksheet calculated
NORMINV (probability, mean, standard_dev)
Returns inverse of normal cumulative distribution for specified mean & standard deviation
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36. Example 10-14 Simulation with Excel
Uniform annual benefit: $250 per year
Initial cost: Normally distributed with mean of $1500 & standard deviation of $150
Useful life: Uniformly distributed over 12 to 16 years
Excel equations:
In B7:B31 Life =12+INT(RAND()*5)
In C7:C31 Initial Cost=NORMINV(RAND(),1500,150)
In D7:D31 IRR=RATE(Life, 250, -Initial Cost)
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37. Simulation Using @Risk
High validity & managerial acceptability
Easier than probability trees
Excel models – uncertain data elements or outcomes
Wide variety of distributions: normal, triangular, beta, …
Discrete & continuous distributions
Simulate with Monte Carlo
Variables can be statistically correlated  good for matching data
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38. Example 10-15 Simulation with @Risk
Simulation easier with programs such & Crystal Ball.
Example to perform 1000 iterations in seconds.
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39. Real Options
Approach to determine value of projects with significant uncertainty
Based on financial options
Values management flexibility in presence of risk
Volatility is key new parameter
Many difficulties exist when applying to real engineering projects
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