engineering economic analysis canadian edition chapter 10: uncertainty in future events

Engineering Economic AnalysisCanadian Edition

Chapter 10:

Uncertainty in Future Events

10-2EECE 450 — Engineering Economics

Chapter 10 … Uses estimated variables to evaluate a

project. Describes uncertain outcomes using

probability distributions. Combines probability distributions of

individual variables for joint probability distributions.

Uses expected values for economic decision making.

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.

Alternative A BCost -$1,000 -$2,000Net annual benefit $150 $250Useful life, in years 10 10Salvage value $100 $400Interest rate 3.50% 3.50% DifferenceNPV $318.38 $362.72 $44.34


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

0 $1000 $2000

1 $400 $700

2 $400 $700

3 $400 $700

4 $400 $700

NPV $267.95 $218.91

Year Project A Project B

0 $1000 $2000

1 $400 $700

2 $400 $700

3 $400 $700

4 $400 $800

NPV $267.95 $287.21


It is good practice to examine the effect on outcomes of variability in the estimates.• 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.

Decision-Making and Uncertainty of Future Outcomes


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:Alternative Optimistic Most likely Pessimistic Mean value

Cost -$950 -$1,000 -$1,150 -$1,016.67Net annual benefit $210 $200 $170 $196.67Useful life, in years 12 10 8 10Salvage value $100 $0 $0 $16.67

IRR = 19.82% 15.10% 3.89% 14.342%Weights: 1 4 1

Mean IRR = 14.016%

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A Range of Estimates … NPV and other criteria are useful for

understanding the impact of future consequences on all sets of scenarios.

If 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 judgment 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.• e.g., project life and annual benefit

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

The expected value is the centre of the probability mass function.

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.

Net Revenue Prob Lifetime Prob$10,000 25% 5 65%$12,800 55% 7 35%$15,000 20% 100%

100%


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

various economic factors (“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:

1. Develop the decision tree.

2. 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.

3. 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 similar to the one below.

Cost (t=0) Expand (t=1) Abandon (t=1)Build: $800,000 $450,000 $600,000Buy: $175,000 $80,000 $0

Build BuyStart net sales: $200,000 $120,000

Build/Continue Build/Expand Buy/Continue Buy/Expand ProbabilityGood net sales: $300,000 $400,000 $160,000 $185,000 60%

Bad net sales: $150,000 $180,000 $96,000 $110,000 40%

MARR: 14%Lifetime (yrs): 10

at t= 0 at t= 1 at t= 1Good= $1,728,548.73 Expand= $1,528,548.73

Continue= $1,483,911.55Bad= $941,955.78 Continue= $741,955.78

Abandon= $600,000.00Build= $440,273.29Buy= $536,392.94

Good= $955,078.79 Expand= $835,078.79Continue= $791,419.49

Bad= $594,851.70 Continue= $474,851.70Abandon= $0.00

Decision is to buy.


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:

1. Formulate the model for determining the project outcome from the project components.

2. Determine the probability distributions of all project components that are random variables.

3. Use a random number generator to produce values for the project components that are random variables and calculate the project outcome using the model.

4. Repeat step 3 until a large enough sample has been taken (usually 150 is a sufficient number).


Simulation … Monte Carlo simulation procedure (cont’d):

5. Produce a frequency distribution and a histogram to estimate the probability distribution of the project outcome.

6. Produce summary statistics of the project outcome, e.g. mean, median, standard deviation, range, minimum, maximum, …

See the examples in the spreadsheet below.

Annual profit Probability$3,500 0.25$5,000 0.50$6,000 0.20$8,000 0.05

Initial cost = $19,000Salvage value = $4,000 $2,171 (PV)

Cost of capital = 13%

NPV mean = $730.71NPV median = $628.24

NPV std. dev. = $1,626.29NPV maximum = $5,351.29NPV minimum = -$3,343.93Prob(NPV < 0) = 35.33%

A machine has an initial cost of $19,000 and an expected lifetime of five years. Its salvage value is expected to be $4,000. The annual profit of operating the machine is uncertain and it can vary from year to year. Its probability distribution is shown below. Analyze the NPV of the project and make a recommendation if the cost of capital is 13 percent.

Simulation Example (Discrete)

NPV Frequency Distribution for Machine

0

5

10

15

20

25

30

35

40

-$3,500 -$2,500 -$1,500 -$500 $500 $1,500 $2,500 $3,500 $4,500 $5,500

NPV


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

engineering economic analysis canadian edition chapter 10: uncertainty in future events

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