1. Chapter 10 Uncertainty in Future Events

2. Chapter Outline Single estimate versus a range of estimates
Probability distributions in economic analysis
Expected value and economic decision trees
Risk versus return
Simulation in economic analysis

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 to describe and solve more complex problems
Measure and consider risk when making economic decisions
Understand how simulation can be used to evaluate economic decisions

4. Estimates in Economic Analysis
Economic analysis requires evaluating the future consequences of an alternative.
Usually, a single value is selected to represent the best estimate that can be made.
Economic analysis was conducted assuming these estimates were correct.

5. Example 10-1 Impact of Estimates in Economic AnalysisB
Cost
$1000
$2000
Net annual benefit
$150
$250
Useful life, in years 10
End-of-useful-life salvage value
$100
$400 ($300)

6. Example 10-2 Use Breakeven in Dealing the Variability in Estimates
Cost
$1000
$2000
Net annual benefit
$150
$250
Useful life, in years 10
End-of-useful-life salvage value
$100 X
Breakeven
Point
For Alternative B to be selected,
NPWB ≥ NPWA
79 + (0.7089)X ≥ 319
X ≥ 339
Copyright Oxford University Press 2009

7. Using a Range of Estimates in Economic Analysis
It is more realistic to describe parameters with a range of possible values.
A range could include an optimistic (O) estimate, the most likely (M) estimate, and a pessimistic (P) estimate.
With Beta distribution, the approximate mean value of a parameter can be calculated as:
(10-1)

8. Example 10-3 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
NPWOptimistic= 0 = (P/A, i, 12) + 100(P/F, i, 12)
IRROptimistic = 19.8%
NPWMost likely = 0 = (P/A, i, 10)
IRRMost Likely = 15.1%
NPWPessimistic = 0 = (P/A, i, 8)
IRRPessimistic = 3.9%

9. Example 10-4 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
Mean Value
$1016.7
$196.7 10
$16.7
NPWMean= 0 = (P/A, i, 10) (P/F, i, 10)
IRRMean = 14.2%

10. Probability It is the likelihood of an event in a single trial.
It also describes the long-run relative frequency of an outcome’s occurrence in many trials.
Probabilities must follow the following rules:
(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.
Continuous distributions: Normal, Continuous uniform, Exponential, and Weibull.
Discrete distributions: Binomial, Uniform, Geometric, Hypergeometric, Poisson, and custom.

11. Example 10-5 Probability 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)

12. Joint Probability Distributions
A joint probability distribution is needed to describe the likelihood of outcomes that combine two or more random variables. Each random variable has its own probability distribution.
If events A and B are independent, the joint probability for both A and B to occur is:
(10-4)

13. 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
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

14. Expected Value
The expected value of a probability distribution is the weighted average of all possible outcomes by their probabilities.
(10-4)
Probabilities can be based on data, expert judgment, or both.
All data in economic analysis may have some level of uncertainty.

15. Example 10-7 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%

16. Example 10-8 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 x 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

17. Example 10-9 Expected Value
Dam Height (ft)
First Cost (I)
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

18. Economic Decision Trees
Economic decision tree graphically displays all decisions in a complex project and all the possible outcomes with their probabilities.
Decision Node D1 D2 DX
Outcome Node
Chance Node C1 C2 CY p1 p2 py
Pruned Branch

19. Economic Decision Trees
7. Net Revenue
=$0
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, …,

20. Example 10-10 Economic Decision Trees
7. Revenue=$0
Terminate
4. Net Revenue
Year 1=$100K
Low Volume
P=0.3
Continue
8.Revenue=$100K/yr
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
6. Net Revenue
Year 1=$400K
Build New
Product No
Continue
10.Revenue=$400K/yr
3. $0
t=0
t=1
t=2, …,

21. Example 10-10 Economic Decision Trees
Build New
Product
2. Volume for
New Product
3. $0 No
Yes
First cost=$1M
4. Net Revenue
Year 1=$100K
7. Revenue=$0
8.Revenue=$100K/yr
6. Net Revenue
Year 1=$400K
9. Revenue=$600K/yr
10.Revenue=$400K/yr
5. Revenue Year 1, 2..8 =$200K
Low Volume
P=0.3
Med. Volume
P=0.6
High Volume
P=0.1
Terminate
Continue
Expand
First cost=$800K
t=0
t=1
t=2, …,
PW1=$550,000
PW=$590,915
PW1=$486,800
EV=$1,046,640
PW=$1,067,000
PW1=$2,120,800
PW=$2,291,660
PW1=$1,947,200

22. Example 10-11 Economic Decision Trees
No accident
P=0.9 $0
EV=$36
Small accident
P=0.07
$300
(<$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
$13,000

23. Risk
Risk can be thought of as the chance of getting an outcome other than the expected value.
Measures of risk:
Probability of a loss (Example 10-6)
Standard deviation ()
(10-6)
(10-7)
(10-7’)

24. Example Risk
Continued from Example 10-11

25. Example 10-13 Risk 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
NPW
Joint
Prob.
-$3,224
0.200
9,842
0.400
18,553
0.067
3,795
0.100
21,072
32,590
0.033
NPW x Joint Prob.
-$645
3,937
1,237
380
4,214
1,086
NPW2 x Joint Prob.
2,079,480
38,747,954
22,950,061
1,442,100
88,797,408
35,392,740
$10,209
=EV(NPW)
$189,409,745
=EV(NPW2)

26. Example 10-14 Risk versus 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% F 3 2 7 6 8 1 5 4

27. Simulation
Simulation is an advanced approach of considering risk in engineering economic analysis.
Economic simulation uses random sampling from the probability distributions of one or more variables to analyze an economic model for many iterations.
For each iteration, all variables with a probability distribution are randomly sampled. These values are used to calculate the NPW, IRR, or EUAW.
The results of all iterations are combined to create a probability distribution for the NPW, IRR, or EUAW.
Simulation can be performed by hand with a table of random number, by using Excel functions, or stand-alone simulation programs such and Crystal Ball.