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Discuss some of the advantages and disadvantages of different approaches to sensitivity analysis, relative to each other.

Lecture slides to accompany
Engineering Economy, 8th edition
Leland Blank, Anthony Tarquin
©McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.
FIN 5203, Finance for Engineers, Trine University
Chapter 18
Sensitivity Analysis and Staged
©McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.
1. Explain sensitivity to parameter variation
2. Use three estimates for sensitivity analysis
3. Calculate expected value E(X)
4. Determine E(X) of cash flow series
5. Use decision trees for staged decisions
6. Understand real options for staged funding
©McGraw-Hill Education.
Parameters and Sensitivity Analysis
Parameter — A variable or factor for which an estimated or stated
value is necessary
Sensitivity analysis – An analysis to determine how a measure
of worth (e.g., PW, AW, ROR, B/C) changes when one or more
parameters vary over a selected range of values.
1. Select parameter to analyze. Assume
independence with other parameters
2. Select probable range and increment
3. Select measure of worth
4. Calculate measure of worth values
5. Interpret results. Graph measure vs.
parameter for better understanding
©McGraw-Hill Education.
Sensitivity of Several Parameters
When several parameters for one alternative vary
and analysis of each parameter is required …
graph percentage change from the most likely estimate
for each parameter vs. measure of worth
Plots with larger slopes
(positive or negative)
have a higher sensitivity
with parameter variation
(sales price curve)
Plots that are relatively
flat have little sensitivity
to parameter variation
(indirect cost curve)
©McGraw-Hill Education.
Three Estimate Sensitivity Analysis
Can be applied to an independent project, or when selecting one
ME alternative from two or more.
Provide a set of parameter estimates for each of the following
three scenarios:
• Pessimistic P
• Most likely ML
• Optimistic O
Calculate measure of worth for each scenario of each alternative,
and select the ‘best’ alternative. This approach takes risk into
account, but there may not be a clear winner.
Note — The pessimistic estimate may be the lowest for some parameters and
the highest for others, e.g., low life estimates and high first cost
estimates are usually pessimistic.
©McGraw-Hill Education.
Example: Three Estimate Sensitivity Analysis
• Without sensitivity analysis would only use the ML row and choose Alternative 1.
• Could ask what alternative would be selected under the pessimistic scenario for
each alternative. Again Alternative 1, which provides added confidence in
choosing Alternative 1.
• A slight concern is the superiority of Alternative 3 under the optimistic scenario,
but Alternative 1 is close behind, shouldn’t change our decision to choose
Alternative 1.
• What if the PW of Alternative 1 is only $500 under the pessimistic scenario?
Then we would most likely select Alternative 3 despite $1,000 lower PW under
the most likely scenario, due to significantly lower downside risk. The expected
value of #3 PW is likely higher that the expected PW of #1 (more on this later).
©McGraw-Hill Education.
Expected Value Calculations
Expected Value — Long-run average observable if a project
or activity is repeated many times
Result is a point estimate based on anticipated outcomes and
estimated probabilities
E ( X ) =  X i P( X i )
i =1
Xi = value of variable X for i = 1, …, m different values
P(Xi) = probability that a specific value of X will occur
In all probability statements, the sum is:
 P( X ) = 1.0
i =1
©McGraw-Hill Education.
When E(X) < 0, e.g., E(PW) = $−2550, a cash outflow is expected; the project is not expected to return the MARR used Example: Probability and Expected Value Monthly M&O cost records over a 4-year period are shown in $200 ranges. Determine the expected monthly cost for next year, if conditions remain constant. Range,$, X No. of months Range,$, X No. of months 100–300 4 700–900 6 300–500 12 900–1100 10 500–700 14 1100–1300 2 Solution: 𝐏(𝐗) = 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐦𝐨𝐧𝐭𝐡𝐬/𝟒𝟖 𝐦𝐨𝐧𝐭𝐡𝐬 𝐄(𝐗) = 𝟐𝟎𝟎(𝟒/𝟒𝟖) + 𝟒𝟎𝟎(𝟏𝟐/𝟒𝟖) + ··· +𝟏𝟐𝟎𝟎(𝟐/𝟒𝟖) = 𝟏/𝟒𝟖[𝟐𝟎𝟎 × 𝟒 + 𝟒𝟎𝟎 × 𝟏𝟐 + ··· +𝟏𝟐𝟎𝟎 × 𝟐] = 𝟏/𝟒𝟖[𝟑𝟏, 𝟐𝟎𝟎] = $𝟔𝟓𝟎/𝐦𝐨𝐧𝐭𝐡 ©McGraw-Hill Education. Expected Value for Alternative Evaluation Two applications for Expected Value for estimates: 1. Prepare information for use in an economic analysis 2. Evaluate economic viability of fully formulated alternative Example: Second use for a complete alternative. Is the investment viable? P = $−5000 n = 3 years MARR = 15% ©McGraw-Hill Education. Example: Expected Value for Alternative Evaluation Solution: Calculate PW value for each condition 𝐏𝐖𝐑 = −𝟓𝟎𝟎𝟎 + 𝟐𝟓𝟎𝟎(𝐏/𝐅, 𝟏𝟓%, 𝟏) + 𝟐𝟎𝟎𝟎(𝐏/𝐅, 𝟏𝟓%, 𝟐) + 𝟏𝟎𝟎𝟎(𝐏/𝐅, 𝟏𝟓%, 𝟑) = $– 𝟔𝟓𝟔 (cash outflow; not viable) 𝐏𝐖𝐒 = $ + 𝟕𝟎𝟖 (cash inflow; viable) 𝐏𝐖𝐄 = $ + 𝟏𝟑𝟎𝟗 (cash inflow; viable) Now, calculate expected value of PW estimates 𝐄(𝐏𝐖) = 𝐏𝐖𝐑 × 𝐏(𝐑) + 𝐏𝐖𝐒 × 𝐏(𝐒) + 𝐏𝐖𝐄 × 𝐏(𝐄) = −𝟔𝟓𝟔 × 𝟎. 𝟒 + 𝟕𝟎𝟖 × 𝟎. 𝟒 + 𝟏𝟑𝟎𝟗 × 𝟎. 𝟐 = $ + 283 On basis of E(PW) > 0 at 15% over 3 years, investment is viable
©McGraw-Hill Education.
Decision Tree Characteristics
Staged Decision – Alternative has multiple stages; decision at one
stage is important to next stage; risk is an inherent element of the
Decision Tree – Helps make risk more explicit for staged decisions
• More than one stage of selection
• Selection of an alternative at one stage
leads to another stage
• Expected results from a decision at each
• Probability estimates for each outcome
• Estimates of economic value (cost or
revenue) for each outcome
• Measure of worth as the selection criterion
©McGraw-Hill Education.
Solving a Decision Tree
Once the tree is developed, probabilities and economic information
are estimated for each outcome branch, and the measure of worth is
selected (usually PW), use the following, starting at top right of tree:
1. Determine PW for each outcome branch
2. Calculate expected value for each alternative:
𝐄(𝐝𝐞𝐜𝐢𝐬𝐢𝐨𝐧) = ∑(𝐨𝐮𝐭𝐜𝐨𝐦𝐞 𝐞𝐬𝐭𝐢𝐦𝐚𝐭𝐞) × 𝐏(𝐨𝐮𝐭𝐜𝐨𝐦𝐞)
3. At each decision node, select the best E(decision) value
4. Continue moving to left to the tree’s root to select the best
5. Trace the best decision path(s) through the tree
©McGraw-Hill Education.
Example: Solving a Decision Tree
1. PW of CFBT is estimated
2. PW for decision nodes
𝐄(𝐢𝐧𝐭’𝐥) = 𝟏𝟐(𝟎. 𝟓) + 𝟏𝟔(𝟎. 𝟓) = 𝟏𝟒
D2 𝐄(𝐧𝐚𝐭’𝐥) = 𝟒(𝟎. 𝟒) − 𝟑(𝟎. 𝟒) − 𝟏(𝟎. 𝟐) = 𝟎. 𝟐
𝐄(𝐢𝐧𝐭’𝐥) = 𝟔(𝟎. 𝟖) − 𝟑(𝟎. 𝟐) = 𝟒. 𝟐
𝐄(𝐧𝐚𝐭’𝐥) = 𝟔(𝟎. 𝟒) − 𝟐(𝟎. 𝟒) + 𝟐(𝟎. 𝟐) = 𝟐
3. Decisions: 14 (int’l) @D2 and 4.2 (int’l)
4. PW for decision node D1
𝐄(𝐦𝐚𝐫𝐤𝐞𝐭) = 𝟏𝟒(𝟎. 𝟐) + 𝟒. 𝟐(𝟎. 𝟖) = 𝟔. 𝟏𝟔
𝐄(𝐬𝐞𝐥𝐥) = 𝟗(𝟏. 𝟎) = 𝟗
Decision: 9 (sell)
5. Trace through tree; select D1 to sell
at E(PW of CFBT ) = $9 million
©McGraw-Hill Education.
Summary of Important Points
Sensitivity analysis evaluates variation in parameters using a specific
measure of worth (PW, ROR, B/C, etc.)
Independence of parameters is assumed in sensitivity analysis
If E(PW) < 0, an alternative is not expected to return the stated MARR, given the estimated probabilities Decision trees assist in making staged decisions when risk is explicitly considered ©McGraw-Hill Education. Purchase answer to see full attachment

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