BENEFITCOST ANALYSIS BENEFITCOST ANALYSIS Financial and Economic Financial and Economic Appraisal using Spreadsheets Appraisal using Spreadsheets
Chapter 9: Risk Analysis
© Harry Campbell & Richard Brown School of Economics The University of Queensland
Risk and Uncertainty
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
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.
Sensitivity Analysis
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.
($ millions at 10% discount rate)
Table 9.1: Sensitivity Analysis Results: NPVs for Hypothetical Road Project
Construction Costs
Road Usage Benefits
75% $50 High Medium $47 $43 Low
100% $40 $36 $32
125% $30 $25 $20
Risk Modeling
Risk modeling is the use of discrete probability distributions to compute expected value of variable rather than point estimate.
Table 9.2: A Discrete Probability Distribution of Road Construction Costs
($ millions)
Road Construction Cost (C) Low Best Guess High
$50 $100 $125
Probability (P) 20% 60% 20%
Table 9.3: Calculating the Expected Value from a Discrete Probability Distribution
($ millions)
20% 60% 20%
E(C)=P x C NPV $86 $10 $36 $60 $11 $25
E(NPV) $17.2 $21.6 $2.2
Road Construction Cost (C) Probability (P) Low $50 Best Guess $100 High $125
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
Joint Probability Distributions
• 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.
Correlated and Uncorrelated Variables
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 $125, and so on.
Table 9.4: Joint Probability Distribution: Correlated Variables
($ millions)
Probability (P) Cost ($)
Benefits ($) Net Benefits ($)
20% 60% 20%
50(10) 100(60) 125(25) (95)
70(14) 125(75) 205(41) (130)
20(4) 25(15) 80(16) (35)
Low Best Guess High (Expected value) Table 9.5: Joint Probability Distribution: Uncorrelated Variables
($ millions)
50 100 125
Probability (P) Probability(P) Cost ($) Benefits ($) Low (L) 20% Best Guess (M) 60% 20% High (H) Joint Probability Combination 0.2 x 0.2 = 0.04 LC-HB 0.2 x 0.6 = 0.12 LC-MB 0.2 x 0.2 = 0.04 LC-LB 0.6 x .0.2 = 0.12 MC-HB 0.6 x 0.6 = 0.36 MC-MB 0.6 x 0.2 = 0.12 MC-LB 0.2 x 0.2 = 0.04 HC-HB 0.2 x 0.6 = 0.12 HC-MB 0.2 x 0.2 = 0.04 HC-LB
70 125 205 Net Benefit ($) 155(6.2) 75(9.0) 20(0.8) 105(12.6) 25(9.0) 30(3.6) 80(3.2) 0(0.0) -55(-2.2) E(NPV) = 42.2
Continuous Probability Distributions
An example is the normal distribution represented as a bell-shaped curve. This distribution is completely described by two parameters:
• the mean • the standard deviation
Degree of dispersion of the possible values around the mean is measured by the variance (s2) or, the square root of the variance – the standard deviation (s).
Frequency (%)
60
40
20
100
-20
20
80
40
60
0
NPV ($ millions)
Figure 9.1: Triangular probability distribution • 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.
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 28
-20
0
40 48
60
80
100
NPV ($ millions)
Figure 9.2: Cumulative Probability Distribution
• The cumulative distribution indicates what the probability is of the NPV lying below (or above) a certain value; • There is a 50% chance that the NPV will be below $28 million, and a 50% chance it will above it; • There is an 80% chance that the NPV will be less than $48 million and a 20% chance that it will more than this.
Using Risk Analysis in Decision Making
Probability
Project A
Project B
A
B
NPV
Figure 9.3: Projects with different degrees of risk
• 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.
R2
R1
G
E(WG)
R0
H
E(WH)
F
E(WF)
D
VAR(W)
VAR(WG)
VAR(WH)
Figure 9.5: A Risk Averse Individual's Indifference Map between Mean and Variance of Wealth
• 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.
with Spreadsheets Using @RISK©
• Add-on for spreadsheet allowing for Monte Carlo simulations; • Instead of entering single point estimate in each input cell, analyst enters information about the probability distribution of variable; • Program then re-calculates NPV or IRR many times over, using a random sample of input data; • Output results (NPVs or IRRs) are then compiled and presented in form of a probability distribution in: - statistical tables - graphical format
