Chapter 10 Uncertainty 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

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

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.

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

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

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)

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 = -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 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 = -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: (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.

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)

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)

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

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.

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%

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

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

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

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, …,

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, …,

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

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

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’)

Example 10-12 Risk Continued from Example 10-11

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)

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

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.