Chapter 10 Uncertainty in Future Events. Chapter Outline Single estimate versus a range of estimates...
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Transcript of Chapter 10 Uncertainty in Future Events. Chapter Outline Single estimate versus a range of estimates...
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
• 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
Learning Objectives
• 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.
Estimates in Economic Analysis
$292
F,3.5%,10)P(300A,3.5%,10)P250(2000NPW B
319$
)10%,5.3,FP(100)10%,5.3,AP(1501000NPWA
A BCost $1000 $2000Net annual benefit $150 $250Useful life, in years 10 10End-of-useful-life salvage value $100 $400 ($300)
363$
)10%,5.3,FP(400)10%,5.3,AP(2502000NPW B
Example 10-1 Impact of Estimates in Economic Analysis
Example 10-2 Use Breakeven in Dealing the Variability in Estimates
$0
$50
$100
$150
$200
$250
$300
$350
$400
$450
0 100 200 300 400 500
X
NP
W
)10%,5.3,FP(X
)10%,5.3,AP(2502000NPW
319$NPW
B
A
A BCost $1000 $2000Net annual benefit $150 $250Useful life, in years 10 10End-of-useful-life salvage value $100 X
Copyright Oxford University Press 2009
For Alternative B to be selected, NPWB ≥ NPWA
79 + (0.7089)X ≥ 319 X ≥ 339
BreakevenPoint
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:
6
PM4OvalueeanM
(10-1)
Example 10-3 Using a Range of Estimates in Economic Analysis
Optimistic Most Likely PessimisticCost $950 $1000 $1150Net annual benefit $210 $200 $170Useful life, in years 12 10 8Salvage value $100 $0 $0
NPWOptimistic= 0 = -950 + 210(P/A, i, 12) + 100(P/F, i, 12)IRROptimistic = 19.8%
NPWMost likely = 0 = -1000 + 200(P/A, i, 10) IRRMost Likely = 15.1%
NPWPessimistic = 0 = -1150 +170(P/A, i, 8) IRRPessimistic = 3.9%
Example 10-4 Using a Range of Estimates in Economic Analysis
Mean Value
$1016.7$196.7
10$16.7
6
PM4O ValueMean
OptimisticMost Likely Pessimistic
Cost $950 $1000 $1150Net annual benefit $210 $200 $170Useful life, in years 12 10 8Salvage value $100 $0 $0
NPWMean= 0 = -1016.7 + 196.7(P/A, i, 10) + 16.7(P/F, i, 10)IRRMean = 14.2%
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:
0yProbabilit1
k to 1j
j 1)P(outcome(10-2)
(10-3)
• Continuous distributions: Normal, Continuous uniform, Exponential, and Weibull.
• Discrete distributions: Binomial, Uniform, Geometric, Hypergeometric, Poisson, and custom.
Example 10-5 Probability
0%
10%
20%
30%
40%
50%
60%
70%
5000 8000 10000
Annual Benefit
Pro
ba
bil
ity
32P(6) ;3
1P(9)
12P(9)P(9)2P(9)P(6) Since
1P(6)P(9)
Optimistic Most Likely PessimisticAnnual benefit $10,000 $8000 $5000Probability P($10,000) 60% 30%Life, in years 9 6Probability P(9) P(6)=2P(9)
0.10.3-0.6-1P($10000)
1)P(outcome j
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:
)B(P)A(PB) andP(A (10-4)
Example 10-6 Joint Probability Distribution
Annual Benefit Probability Life Probability$5,000 0.3 6 0.67
8,000 0.6 6 0.6710,000 0.1 6 0.67
5,000 0.3 9 0.338,000 0.6 9 0.33
10,000 0.1 9 0.33
0.0%
10.0%
20.0%
30.0%
40.0%
50.0%
($3,224) $9,842 $18,553 $3,795 $21,072 $32,590
NPW
Pro
bab
ility
NPWJoint
Probability-$3,224 0.200
9,842 0.40018,553 0.067
3,795 0.10021,072 0.20032,590 0.033
Expected Value
• The expected value of a probability distribution is the weighted average of all possible outcomes by their probabilities.
P(B)OutcomeP(A)Outcome
P(j)OutcomeValueExpected
BA
jallj
(10-4)
Example 10-7 Expected Value
105360%,7)7300(P/A,125000NPWEV
73005000(0.3)8000(0.6)10000(0.1)EVBenefit
Optimistic Most Likely PessimisticAnnual benefit $10,000 $8000 $5000Probability 10% 60% 30%Life, in years 9 6Probability 33.3% 66.7%
76(0.667)9(0.333)EVLife
Example 10-8 Joint Probability Distribution
Annual Benefit Probability Life Probability$5,000 0.3 6 0.67
8,000 0.6 6 0.6710,000 0.1 6 0.67
5,000 0.3 9 0.338,000 0.6 9 0.33
10,000 0.1 9 0.33
PWJoint
ProbabilityPW x JointProbability
-$3,224 0.200 -$6459,842 0.400 3,937
18,553 0.067 1,2373,795 0.100 380
21,072 0.200 4,21432,590 0.033 1,086
EV(PW)=10,209
Example 10-9 Expected Value
Dam Height (ft)EUAC of First Cost
Expected Annual Flood Damage
Total Expected EUAC
No dam $0 $200,000 $200,00020 38,344 25,000 63,34430 43,821 3,000 46,82140 49,299 400 49,699
50) 5%, I(A/P,EUAC
Dam Height (ft)First Cost
(I)Annual
P(Flood>Height)Damage if
Flood OccursNo dam $0 0.25 $800,000
20 700,000 0.05 500,00030 800,000 0.01 300,00040 900,000 0.002 200,000
Height)P(FloodDamageEV Damage Annual
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
Chance Node
C1
C2
CY
p1
p2
py
Outcome Node
Pruned Branch
Economic Decision Trees
1. Build New Product
2. Volume forNew Product
3. $0No
YesFirst cost=$1M
4. Net Revenue Year 1=$100K
7. Net Revenue =$0
8. Net Revenue $100K/year
6. Net Revenue Year 1=$400K
9. Net Revenue =$600K/year
10. Net Revenue =$400K/year
5. Net Revenue Year 1=$200K Year 2…n=$200K
Low Volume P=0.3
Med. Volume P=0.6
High Volume P=0.1
Terminate
Continue
Continue
t=0 t=1 t=2, …,
ExpandFirst cost=$800KExpandFirst cost=$800K
Example 10-10Economic Decision Trees
1. Build New Product
2. Volume forNew Product
3. $0No
YesFirst 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
Continue
ExpandFirst cost=$800K
t=0 t=1 t=2, …,
Example 10-10Economic Decision Trees
1. Build New Product
2. Volume forNew Product
3. $0No
YesFirst 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
Continue
ExpandFirst cost=$800K
t=0 t=1 t=2, …,
PW1=$550,000
PW1=$486,800PW=$590,915
PW=$1,067,000
PW1=$2,120,800
PW1=$1,947,200PW=$2,291,660
EV=$1,046,640
Example 10-11Economic Decision Trees
$0
$300 (<$500 deductible)
$500TotaledP=0.03
$0
$300
$13,000
No accidentP=0.9
Small accidentP=0.07
TotaledP=0.03
Buy Insurance$800
Self-Insure$0
EV=$36
EV=$411
No accidentP=0.9No accidentP=0.9
Small accidentP=0.07
Small accidentP=0.07
Risk
• Risk can be thought of as the chance of getting an outcome other than the expected value.
• Measures of risk:• Probability of a loss (Example 10-6)• Standard deviation ()
2
j
2j [EV(X)]-P(j)Outcome
22 [EV(X)]-)EV(X
(10-6)
(10-7)
(10-7’)
]mean)-[EV(X 2
Example 10-12 Risk
411$)03.0(13000$)07.0(300$)9.0(0$EV
36$)03.0(500$)07.0(300$)9.0(0$EV
InsureSelf
InsuranceBuy
2215$411-0763005][EV-EV
112$36-38001][EV-EV
22InsureSelf
2InsureSelfInsureSelf
22InsuranceBuy
2InsuranceBuyInsuranceBuy
300,076,5)03.0(13000$)07.0(300$)9.0(0$EV
800,13)03.0(500$)07.0(300$)9.0(0$EV2222
InsureSelf
2222InsuranceBuy
Continued from Example 10-11
Example 10-13 Risk
Annual Benefit Prob.
Life (years) Prob.
$5,000 0.3 6 0.678,000 0.6 6 0.67
10,000 0.1 6 0.675,000 0.3 9 0.338,000 0.6 9 0.33
10,000 0.1 9 0.33
9229$10,209-457,409,189][EV-EV 22NPW
2NPWNPW
NPWJointProb.
-$3,224 0.2009,842 0.400
18,553 0.0673,795 0.100
21,072 0.20032,590 0.033
NPW x Joint Prob.
-$6453,9371,237
3804,2141,086
$10,209=EV(NPW)
$189,409,745=EV(NPW2)
NPW2 x Joint Prob.
2,079,48038,747,95422,950,061
1,442,10088,797,40835,392,740
Example 10-14 Risk versus Returns
0%
5%
10%
15%
20%
0% 2% 4% 6% 8% 10%
Standard Deviation of IRR
Ex
pe
cte
d V
alu
e o
f IR
R
Project IRR Std. Dev.
1 13.10% 6.50%
2 12.00% 3.90%
3 7.50% 1.50%
4 6.50% 3.50%
5 9.40% 8.00%
6 16.30% 10.00%
7 15.10% 7.00%
8 15.30% 9.40%
F 4.00% 0.00%
F
3
2
76
8
1
5
4
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 as @Risk and Crystal Ball.