Chapter 10Uncertainty in Future Events

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.