Engineering Economic Analysis Canadian Edition Chapter 10: Uncertainty in Future Events

Chapter 10 … Uses the expected values of variables to evaluate projects and make decisions. Describes uncertain outcomes using probability distributions. Combines the probability distributions of single variables in joint probability distributions. Measures/assesses risk in decision making. Uses simulations for decision making.

Precise estimates are still estimates All estimates are inherently uncertain. Far-term estimates are almost always more uncertain than near-term estimates. Minor changes in any estimate(s) may alter the results of an economic analysis. Using breakeven and sensitivity analysis yields an understanding of how changes in variables will affect the economic analysis.

Decision-Making and Uncertainty of Future Outcomes In the left box, one cash flow in Project B is uncertain and NPVA > NPVB (i = 10%). In right box, the cash flow estimate has changed and now NPVA < NPVB. Year Project A Project B $1000 $2000 1 $400 $700 2 3 4 NPV $267.95 $218.91 Year Project A Project B $1000 $2000 1 $400 $700 2 3 4 $800 NPV $267.95 $287.21

Decision-Making and Uncertainty of Future Outcomes … It is good practice to examine the effects of variability in the estimates on the outcomes. By how much and in what direction will a measure of merit (e.g., NPV, EACF, IRR) be affected by variability in the estimates? But, this does not take the inherent variability of parameters into account in an economic analysis. We need to consider a range of estimates.

A Range of Estimates Usually, we consider three scenarios: Optimistic Most likely Pessimistic Compute the internal rate of return for each scenario. Compare each IRR to the MARR. What if IRR < MARR for one or more scenarios? Are scenarios equally important? Need a weighting scheme.

A Range of Estimates … Assigning weights Assign a weight to each of the three scenarios. Usually, the largest weight goes to the most likely scenario. The optimistic and pessimistic scenarios may have equal or unequal weights. There are two possible approaches: Calculate the weighted average of the measure of merit (e.g., IRR) for all scenarios. Calculate the measure of merit (e.g., IRR) from the weighted average of each parameter (annual benefit and cost; first cost; salvage value …).

A Range of Estimates … There is often a range of possible values for a parameter instead of a single value:

A Range of Estimates … NPV and other criteria are useful for understanding the impact of future consequences on all sets of scenarios. Suppose several variables are uncertain. It is unlikely that all variables will be optimistic, pessimistic, or most likely. Calculate weighted average values for each parameter based on the scenario weights. Use the preceding example to explore the effect on the IRRs of increasing or decreasing the variability of the parameters.

Probability and Risk Probabilities of future events can be based on data, judgement, or a combination of both. Weather and climate data; expert judgement on events. Most data has some level of uncertainty. Small uncertainties are often ignored. Variables can be known with certainty (deterministic) or with uncertainty (random or stochastic).

Probability and Risk … 0 ≤ Probability ≤ 1 The sum of probabilities for all possible outcomes = 1 or 100%. It is usual in engineering economics to use between two and five outcomes with discrete probabilities. Expert judgement limits the number of outcomes. Each additional outcome requires more analysis. Probability can be considered as the long-run relative frequency of an outcome’s occurrence.

Joint Probability Distributions Random variables are assumed to be statistically independent; for instance, the project lifetime and the annual benefit are assumed to be independent (unrelated). Project criteria, e.g., NPV, IRR, depend on the probability distributions of input variables. We need to determine the joint probability distributions of different combinations of input parameters.

Joint Probability Distributions … If A and B are independent, P(A and B) = P(A)  P(B); “A and B” means that A and B occur simultaneously (intersection). Suppose there are three values for the annual benefit and two values for the life. This leads to six possible combinations that represent the full set of outcomes and probabilities. Joint probability distributions are burdensome to construct when there is a large number of variables or outcomes.

Expected Value The expected value is the mean of the random variable using the values of the variable and their probabilities to calculate a weighted average. E[X] = μ = (pj)(xj) for all j p = probability; x = discrete value of the variable An expected value can be determined when two or more possible outcomes and their associated probabilities are known.

Expected Value … Example: a firm is considering an investment that has annual net revenue and lifetime (in years) with the probability distributions shown below. Find the joint probability distribution. Find the expected value of the NPV by using the expected values of the parameters, then by finding the expected value of the NPVs.

Decision Tree Analysis A decision tree is a logical structure of a problem in terms of the sequence of decisions and outcomes of chance events. e.g. demand for a new product will depend on economic factors or conditions (“states of nature”). Decisions depending on the outcomes of random events force decision makers to anticipate what those outcomes might be as part of the analysis process. This analysis is suited to decisions and events that have a natural sequence in time or space.

Decision Tree Analysis … A decision tree grows from left to right and usually begins with a decision node. Represents a decision required by the decision maker. Branches extending from a decision node represent decision options available to the decision maker. A chance node represents events for which outcomes are uncertain. Branches extending from a chance node represent possible outcome factors, sometimes called “states of nature”.

Decision Tree Analysis … The decision tree analysis procedure: Develop the decision tree. Execute the rollback procedure on the decision tree from right to left. Compute the expected value (EV) of each possible outcome at each chance node. Select the option with best EV. Continue the rollback process until the leftmost node is reached. Select the expected value associated with the final node.

Decision Tree Analysis … Example: the manufacturing engineers of a firm want to decide whether the company should build its new product or have it built under contract (the “build or buy” decision). The initial cost of building the product is $800,000 and net sales will be $200,000 in the first year. The initial cost of buying the product is $175,000 and net sales will be $120,000 in the first year. After one year of operation, the company has the choice either to continue with the product, expand operations, or abandon the product.

Decision Tree Analysis … The decision will depend on whether the economy is good or bad. The probability of a good economy in one year is 60%. If they build the product, in one year: the cost of expanding will be $450,000; they could receive $600,000 if they abandon; if they continue, net annual sales will be $300,000 if the economy is good and $150,000 if it is bad; if they expand, net annual sales will be $400,000 if the economy is good and $180,000 if it is bad.

Decision Tree Analysis … If they buy the product, in one year: the cost of expanding will be $80,000; they will receive $0 if they abandon; if they continue, net annual sales will be $160,000 if the economy is good and $96,000 if it is bad; if they expand, net annual sales will be $185,000 if the economy is good and $110,000 if it is bad. They use a MARR of 14% for decisions like this and the expected lifetime is 10 years. Perform a decision tree analysis and make a recommendation to the engineers.

Decision Tree Analysis … The solution can be worked out by hand with the aid of a decision tree, and it can be verified using a spreadsheet, as below.

Simulation When any of the project components is a random variable, the outcome of the project, e.g. the NPV, is also a random variable. If we want to assess a project with uncertain parameters, we estimate the probability distribution of the outcome using the relative frequency approach. We can do this by repeatedly sampling from the distributions of the project’s parameters. Spreadsheets are helpful in this procedure.

Simulation … Monte Carlo simulation procedure: Formulate the model for determining the project outcome from the project components. Determine the probability distributions of all project components that are random variables. Use a random number generator to produce values for the project components that are random variables and calculate the project outcome using the model. Repeat step 3 until a large enough sample has been taken (250 is usually a sufficient number).

Simulation … Monte Carlo simulation procedure (cont’d): Produce a frequency distribution and a histogram to estimate the probability distribution of the project outcome. Produce summary statistics of the project outcome, e.g. mean, median, standard deviation, range, minimum, maximum, …

Simulation … Example: A machine initially costs $19,000 and has an expected lifetime of five years. Its salvage value is expected to be $4000. The annual profit from operating the machine has the probability distribution shown in the table below. Perform a simulation of the NPV of the project and make a recommendation for a cost of capital of 13 percent. See the spreadsheet on the second page following. Annual profit $3500 $5000 $6000 $8000 Probability 0.25 0.50 0.20 0.05

Simulation … Example: A machine’s initial cost is uniformly distributed between $18,000 and $20,000 and its salvage value is expected to be $4000. The annual profit of operating the machine has a normal distribution with a mean of $6000 and a standard deviation of $2500. The machine’s expected lifetime is five years. Use simulation to analyze the project for a cost of capital of 13%. A spreadsheet for both discrete and continuous distributions is on the following page (click on the tabs when it is open).

Simulation …

Suggested Problems 10-15, 17, 18, 20, 22, 23, 24, 25, 28, 31.