© Harry Campbell & Richard Brown School of Economics The University of Queensland BENEFIT-COST ANALYSIS BENEFIT-COST ANALYSIS Financial and Economic Financial and Economic Appraisal using Spreadsheets Appraisal using Spreadsheets Chapter 9: Risk Analysis In the preceding chapters we assumed all costs and benefits are known with certainty. The future is uncertain: • factors internal to the project • factors external to the project Risk and Uncertainty Where the possible values could have significant impact on project’s profitability, a decision will involve taking a risk. In some situations, degree of risk can be objectively determined. Estimating probability of an event usually involves subjectivity. Risk and Uncertainty In risk analysis different forms of subjectivity need to be addressed in deciding: • what the degree of uncertainty is; • whether the uncertainty constitutes a significant risk; • whether the risk is acceptable. Establishing the extent to which the outcome is sensitive to the assumed values of the inputs: • it tells how sensitive the outcome is to changes in input values; • it doesn’t tell us what the likelihood of an outcome is. Sensitivity Analysis Table 9.1: Sensitivity Analysis Results: NPVs for Hypothetical Road Project ($ millions at 10% discount rate) Construction Costs 75% 100% 125% High $50 $40 $30 Medium $47 $36 $25 Road Usage Benefits Low $43 $32 $20 Risk modeling is the use of discrete probability distributions to compute expected value of variable rather than point estimate. Risk Modeling Table 9.3: Calculating the Expected Value from a Discrete Probability Distribution ($ millions) Road Construction Cost (C) Probability (P) E(C)=P x C NPV E(NPV) Low $50 20% $10 $86 $17.2 Best Guess $100 60% $60 $36 $21.6 High $125 20% $25 $11 $2.2 The expected cost of road construction can be derived as: E(C) = $10 + $60 + $25 = $95 And the expected NPV as: E(NPV) = 17.2 + 21.6 + 2.2 = $41 Table 9.2: A Discrete Probability Distribution of Road Construction Costs ($ millions) Road Construction Cost (C) Probability (P) Low $50 20% Best Guess $100 60% High $125 20% • Usually uncertainty about more than one input or output; • The probability distribution for NPV depends on aggregation of probability distributions for individual variables; • Joint probability distributions for correlated and uncorrelated variables. Joint Probability Distributions Assume that if road usage increases, so to do road maintenance costs. There is a 20% chance of road maintenance costs being $50 and road user benefits being $70; a 60% chance of road maintenance costs being $100 and road user benefits being $125, and so on. Correlated and Uncorrelated Variables Table 9.4: Joint Probability Distribution: Correlated Variables ($ millions) Probability (P) Cost ($) Benefits ($) Net Benefits ($) Low 20% 50(10) 70(14) 20(4) Best Guess 60% 100(60) 125(75) 25(15) High 20% 125(25) 205(41) 80(16) (Expected value) (95) (130) (35) Table 9.5: Joint Probability Distribution: Uncorrelated Variables ($ millions) Probability (P) Probability(P) Cost ($) Benefits ($) Low (L) 20% 50 70 Best Guess (M) 60% 100 125 High (H) 20% 125 205 Combination Joint Probability Net Benefit ($) LC-HB 0.2 x 0.2 = 0.04 155(6.2) LC-MB 0.2 x 0.6 = 0.12 75(9.0) LC-LB 0.2 x 0.2 = 0.04 20(0.8) MC-HB 0.6 x .0.2 = 0.12 105(12.6) MC-MB 0.6 x 0.6 = 0.36 25(9.0) MC-LB 0.6 x 0.2 = 0.12 30(3.6) HC-HB 0.2 x 0.2 = 0.04 80(3.2) HC-MB 0.2 x 0.6 = 0.12 0(0.0) HC-LB 0.2 x 0.2 = 0.04 -55(-2.2) E(NPV) = 42.2 An example is the normal distribution represented as a bell-shaped curve. This distribution is completely described by two parameters: • the mean • the standard deviation Degree of dispersion of the possible values around the mean is measured by the variance (s 2 ) or, the square root of the variance – the standard deviation (s). Continuous Probability Distributions [...]... it will more than this Using Risk Analysis in Decision Making Figure 9. 3: Projects with different degrees of risk Probability Project A Project B A B NPV • Choice depends on decision-maker’s attitude towards risk; • B has higher expected NPV, but is riskier than A; • final choice depends on how much the decision-maker is risk averse or is a risk taker Figure 9. 5: A Risk Averse Individual's Indifference...Figure 9. 1: Triangular probability distribution Frequency (%) 60 40 20 -20 0 20 40 60 80 100 NPV ($ millions) • triangular or ‘three-point’ distribution offers a more formal risk modeling exercise than a sensitivity analysis; • the distribution is described by a high (H), low (L) and best-guess (B) estimate; • provide the maximum, minimum and modal values of the distribution respectively Figure 9. 2:... R0 H F D VAR(WH) VAR(WG) VAR(W) • Shape of indifference map shows how the decision-maker perceives risk; • Slope shows amount by which E(W) needs to increase to offset any given increase in risk; • The larger this amount is, the more risk averse the individual is at the given level of wealth Using @RISK with Spreadsheets © • Add-on for spreadsheet allowing for Monte Carlo simulations; • Instead of... high (H), low (L) and best-guess (B) estimate; • provide the maximum, minimum and modal values of the distribution respectively Figure 9. 2: Cumulative Probability Distribution Cumulative Frequency 1.0 0 .9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -20 0 20 28 40 48 60 80 100 NPV ($ millions) • The cumulative distribution indicates what the probability is of the NPV lying below (or above) a certain value; • There . Queensland BENEFIT-COST ANALYSIS BENEFIT-COST ANALYSIS Financial and Economic Financial and Economic Appraisal using Spreadsheets Appraisal using Spreadsheets Chapter 9: Risk Analysis In the preceding chapters. attitude towards risk; • B has higher expected NPV, but is riskier than A; • final choice depends on how much the decision-maker is risk averse or is a risk taker. Figure 9. 5: A Risk Averse Individual's. will more than this. Figure 9. 3: Projects with different degrees of risk NPV B A Project B Project A Probability Using Risk Analysis in Decision Making •