Download Chapter 10 Engineering Economic Analysis (Canadian Edition)

Engineering Economic Analysis
Canadian Edition
Chapter 10:
Uncertainty in Future Events
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.
10-2
Precise estimates are still estimates
 All estimates are inherently uncertain and farterm 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
Cost
Net annual benefit
Useful life, in years
Salvage value
Interest rate
NPV
A
-$1,000
$150
10
$100
3.50%
$318.38
B
-$2,000
$250
10
$400
3.50% Difference
$362.72
$44.34
10-3
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
Year
Project A
Project B
0
$1000
$2000
0
$1000
$2000
1
$400
$700
1
$400
$700
2
$400
$700
2
$400
$700
3
$400
$700
3
$400
$700
4
$400
$700
4
$400
$800
NPV
$267.95
$218.91
NPV
$267.95
$287.21
10-4
Decision-Making and Uncertainty of
Future Outcomes
 It is good practice to examine the variability of
estimates on 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.
10-5
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.
10-6
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 …).
10-7
A Range of Estimates …
 There is often a range of possible values for a
parameter instead of a single value:
Alternative
Cost
Net annual benefit
Useful life, in years
Salvage value
IRR =
Weights:
Mean value =
Optimistic Most likely Pessimistic Mean value
-$950
-$1,000
-$1,150
-$1,016.67
$210
$200
$170
$196.67
12
10
8
10
$100
$0
$0
$16.67
19.82%
15.10%
3.89%
14.342%
1
4
1
14.016%
w1O  w 2ML  w 3P
Expected value : E ( x ) 
w1  w 2  w 3
10-8
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.
10-9
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).
10-10
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.
10-11
Probability and Risk …
 Examples of Games of Chance
Value
1st roll
2nd in a row
2
2.7778%
0.0772%
3
5.5556%
0.3086%
4
8.3333%
0.6944%
5
11.1111%
1.2346%
6
13.8889%
1.9290%
7
16.6667%
2.7778%
8
13.8889%
1.9290%
9
11.1111%
1.2346%
10
8.3333%
0.6944%
11
5.5556%
0.3086%
12
2.7778%
0.0772%
Sum 100.0000% 11.2654%
Total "n" items Probability
52
1 1.9231%
52
4 7.6923%
52
16 30.7692%
52
26 50.0000%
10-12
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.
10-13
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.
10-14
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.
10-15
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 and by
finding the expected value of the NPVs.
Net Revenue
$10,000
$12,800
$15,000
Prob
Lifetime Prob
25%
5 65%
55%
7 35%
20%
100%
100%
10-16
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.
10-17
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”.
10-18
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.
10-19
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.
10-20
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.
10-21
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.
10-22
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.
Build:
Buy:
Cost (t=0)
$800,000
$175,000
Start net sales:
Build
$200,000
Good net sales:
Bad net sales:
Expand (t=1) Abandon (t=1)
$450,000
$600,000
$80,000
$0
Buy
$120,000
Build/Continue Build/Expand Buy/Continue Buy/Expand
$300,000
$400,000
$160,000
$185,000
$150,000
$180,000
$96,000
$110,000
MARR:
Lifetime (yrs):
14%
10
at t= 0
Build=
Buy=
Probability
60%
40%
$440,273.29
$536,392.94
Good=
at t= 1
$1,728,548.73
Bad=
$941,955.78
at t= 1
Expand= $1,528,548.73
Continue= $1,483,911.55
Continue=
$741,955.78
Abandon=
$600,000.00
Decision is to buy.
Good=
$955,078.79
Bad=
$594,851.70
Expand=
Continue=
Continue=
Abandon=
$835,078.79
$791,419.49
$474,851.70
$0.00
10-23
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.
10-24
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).
10-25
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.
Simulation Example (Discrete)
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.
Annual profit
$3,500
$5,000
$6,000
$8,000
Probability
0.25
0.50
0.20
0.05
Initial cost =
Salvage value =
Cost of capital =
$19,000
$4,000
13%
NPV mean =
NPV median =
NPV std. dev. =
NPV maximum =
NPV minimum =
Prob(NPV < 0) =
$730.71
$628.24
$1,626.29
$5,351.29
-$3,343.93
35.33%
NPV Frequency Distribution for Machine
40
35
30
25
$2,171 (PV)
20
15
10
5
0
-$3,500 -$2,500 -$1,500 -$500
$500
$1,500 $2,500 $3,500 $4,500 $5,500
NPV
10-26
Suggested Problems
 10-15, 17, 18, 20, 22, 23, 24, 25, 28, 31.
10-27