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 Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2017

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? Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2014 Copyright Oxford University Press 2017

Why Do Sensitivity Analysis? Data is uncertain Make better decisions Decide which data merits refinement Focus managerial attention during implementation All of the above Copyright Oxford University Press 2017

Why Do Sensitivity Analysis? Data is uncertain Make better decisions Decide which data merits refinement Focus managerial attention during implementation All of the above Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2017

Mean Value from a Range To calculate a mean or average from a range Approximation using beta distribution Widely used in project management Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2017

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. Copyright Oxford University Press 2017

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 = 1 3 ; 𝑃 6 = 2 3 Copyright Oxford University Press 2017

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) Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2017

Expected Value Expected value = weighted average of outcomes times their probabilities (10-5) Copyright Oxford University Press 2017

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% Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2017

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) Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2017

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, …, Copyright Oxford University Press 2017

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, …, Copyright Oxford University Press 2017

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, …, Copyright Oxford University Press 2017

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 = 800 + 36 = $836 Self-insure = $411 $13,000 Copyright Oxford University Press 2017

Risk Risk is chance of getting an outcome other than expected value Measures of risk Probability of a loss Standard deviation (σ) Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2017

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) = Copyright Oxford University Press 2017

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% Copyright Oxford University Press 2017

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 as @Risk or Crystal Ball Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2017

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) Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2017

Example 10-15 Simulation with @Risk Simulation easier with programs such as @Risk & Crystal Ball. Example uses @Risk to perform 1000 iterations in seconds. Copyright Oxford University Press 2017

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 Copyright Oxford University Press 2017