ᎏᎏ X CA/U ⫻ X CO/EXCA/E/XCA/U
6.10 Decision-Making Under Risk, Sensitivity, and Uncertainty
Economic analysis is usually concerned with the present and future consequences of investment alternatives.
However, estimating the consequence of future costs and benefits is not always easy, but it must be done for decision purposes. The accuracy of such estimates is an important element of the results of economic analy- ses. When the degree of confidence on data, information, and techniques used in estimating future cash flows is 100%, then this kind of analysis is called decisions under certainty. However, there is hardly any sit- uation in which the confidence level is 100%; there always exist some elements of uncertainty attached to every decision. The uncertainty surrounding cash-flow data for project evaluation is multidimensional in nature, and the vagueness of one factor interplays with the vagueness of the other factors to create an even more complicated decision-making scenario. There are several techniques for handling uncertainty in eco- nomic analysis and they can be classified into probabilistic, nonprobabilistic, and fuzzy techniques.
Both risk and uncertainty in cash-flow estimates are caused by a lack of accurate data and information regarding the future conditions of the investments under consideration. Such future conditions may include changes in technology, the relationship between independent projects, the impacts of international
trades and businesses, and several other peculiar conditions. The terms “risk” and “uncertainty” are used interchangeably in economic analysis. However, decision under risk involves situations in which the future estimates can be estimated in terms of probability of occurrences. Decision under uncertainty, on the other hand, involves situations in which such future estimates cannot be estimated in terms of probability of occurrences. Therefore, it is usually helpful to determine the impact of a change in a cash-flow estimate on the overall capital investment decision; that is, to determine how sensitive an investment is to changes in a given cash-flow estimate that is not known with certainty.
6.10.1 Sources of Uncertainty in Project Cash Flows
Some of the aspects of investment cash flows that contain imprecision include the economic life of a proj- ect, cash-flow estimates, estimating the MARR, estimating the effects of inflation, and timing of cash flows. The economic life of a project may not be known with certainty because the life of the technology used in developing the project may not be known with certainty either. In most cases, the technology may be new and there may not be enough information to make a probabilistic approximation. In addition, some of the technology may be shorter lived than initially anticipated. Cash-flow estimates for each proj- ect phase are a difficult problem. Costs and benefits streams for projects are usually indefinite. In addi- tion, the benefits may also be delayed as a result of project extension and unexpected additional project costs. The MARR for projects is usually project-dependent; however, the method of determining project MARR is better modeled with uncertainty because there may be a delay in accruing project benefits, the project costs may exceed budgeted values, the benefits may not be as huge as initially thought, or the tech- nology may be shorter lived than initially anticipated. Inflation becomes a major concern in projects that take several years to complete. Therefore, estimating the impact of inflation is not possible with certainty.
Shifted cash flows are another issue when projects are not completed on schedule and may have some ele- ments of probability attached to it. A reasonable probabilistic assumption of how long the project will take may help in accounting for shifted cash flows.
6.10.2 Non-Probabilistic Models
There are several nonprobabilistic techniques, such as breakeven analysis, sensitivity graph (spiderplot), and the use of a combination of these two factors (Sullivan et al., 2003). However, breakeven analysis remains the most popular.
Breakeven analysis is used when the decision about a project is very sensitive to a single factor and this factor is not known with certainty. Therefore, the objective of this technique is to determine the breakeven point (QBE) for this decision variable. The approach can be used for a single project or for two projects. The breakeven technique usually assumes a linear revenue relation, but a nonlinear relation is often more realistic. When one of the parameters of an evaluating technique, such as P, F, A, I, or n for a single project, is not known or not estimated with certainty, a breakeven technique can be used by setting the equivalent relation for the PW, AW, ROR, or B/C equal to 0 in order to determine the breakeven point for the unknown parameter. The project may also be modeled in terms of its total revenue and total cost (fixed cost plus variable cost). Therefore, at some unit of product quantity, the revenue and the total cost relations intersect to identify the breakeven quantity. This identified quantity is an excellent starting tar- get for planning purposes. Product quantity less than the breakeven quantity indicates a loss; while prod- uct quantity greater than the breakeven quantity indicates a profit.Figure 6.10 shows linear and nonlinear breakeven graphs for a single project.
The breakeven technique can also be used to determine the common economic parameters between two competing projects. Some of the parameters that may be involved are the interest rate, the first cost, the annual operating cost, the useful life, the salvage value, and the ROR, among others. The steps used in this case can be summarized as follows:
1. Define the parameter of interest and its dimension.
2. Compute the PW or AW equation for each alternative as a function of the parameter of interest.
3. Equate the two equations and solve for the breakeven value of the parameter of interest.
4. If the anticipated value is above this calculated breakeven value, select the alternative with the lower parameter cost (smaller slope). If otherwise, select the alternative with the higher parameter cost (larger slope). Figure 6.11 shows a graphical example.
This approach can also be used for three or more alternatives by comparing the alternatives in pairs to find their respective breakeven points. Figure 6.12 shows a graphical example for three alternatives.
$
QBE Q units per year (a)
Total cost
Variable cost $
QBE Q units per year Nonlinear cost relations Linear cost relations
Total cost Variable cost
(b) FIGURE 6.10 Linear and nonlinear breakeven graphs.
QBE
Alt. 2 total cost Alt. 1 total cost
Total cost
FIGURE 6.11 Breakeven between two alternatives with linear relations.
..
Total cost
Q13 Q12 Q23
Alt. 2 total cost Alt. 1 total cost
Alt. 3 total cost
FIGURE 6.12 Breakeven between three alternatives with linear relations.
6.10.3 Probabilistic Models
A probabilistic model is decision making under risk and involves the use of statistics and probability. The most popular is Monte Carlo sampling and simulation analysis.
The simulation approach to engineering economic analysis is summarized as follows (Blank and Tarquin, 2002):
1. Formulate alternative(s) and select the measure of worth to be used.
2. Select the parameters in each alternative to be treated as random variables and estimate values for other definite parameters.
3. Determine whether each variable is discrete or continuous and describe a probability distribution for each variable in each alternative.
4. Develop random samples.
5. Compute n values of the selected measure of worth from the relation(s) in step 1 using the defi- nite estimates made and n sample values for the varying parameters.
6. Construct the probability distribution of the measure computed in step 5 using between 10 and 20 cells of data and calculate measures such as the mean, the root-mean-square deviation, and other relevant probabilities.
7. Draw conclusions about each alternative and decide which is to be selected.
The results of this approach can be compared with decision-making when parameter estimates are made with certainty.