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CHAPTER 10
Dealing With Uncertainty
RISK
• Risk and uncertainty are similar
• Both present the problem of not knowing what
future conditions will be
• Risk offers estimates of probabilities for possible
outcomes
• Uncertainty does not provide estimates of
probabilities for possible outcomes
• This book treats them as interchangeable
Four Major Sources Of Uncertainty
1. Possible inaccuracy of cash-flow estimates used in the
study
–
–
How much source information is available
How dependable is the source information
2. Type of business relative to the future health of the
economy
–
Some businesses will typically be more at risk when there is
a general decline in the economy
3. Type of physical plant and equipment involved
–
Some equipment have more definite lives and MV than the
others (general lathe machine vs. mining equipment)
4. Length of study period
–
The longer the study period, the greater the level of
uncertainty of a capital investment
Sensitivity Analysis
•
•
Sensitivity – The degree to which a measure of
merit (I.e., PW, IRR, etc…) will change as a
result of changes in one or more of the study
factor values
Sensitivity Analysis Techniques
1. Breakeven Analysis
2. Sensitivity Graph (spiderplot)
3. Combination of factors
Breakeven Analysis
• Useful for choosing among alternatives when costs or
revenues are highly sensitive to a single factor that is
hard to estimate (e.g., operating hours per year, useful
life, etc.)
• General Procedure:
– Write an expression of equivalent worth for each alternative in
terms of the common factor.
– Equate the equivalent worths and solve for the value of the
common factor. This value is the breakeven point (B.E.P.).
– Estimate whether the actual factor value will be higher or
lower than the B.E.P. and then choose the appropriate
alternative.
Breakeven Problem Involving Two
Alternatives
•
•
Indifference between alternatives (EWA = f1(y);
EWB = f2(y)
EWA = EWB; f1(y) = f2(y) : Solve for y
Economic acceptability of engineering project
EWp = f(z) = 0
The value of ‘z’ is the value at which we would
be indifferent between accepting or rejecting
the project
Example
• Suppose that there are two alternative electric motors that provide
100 hp output. An Alpha motor can be purchased for $12,500 and
has an efficiency of 74%, an estimated life of 10 years, and
estimated maintenance cost of $500 per year. A Beta motor will
cost $16,000 and has an efficiency of 92%, a life of 10 years, and
annual maintenance costs of $250. Annual taxes and insurance
costs on either motor will be 1-1/2% of the investment. If the
minimum attractive rate of return is 15%, how many hours per
year would the motors have to be operated at full load for the
annual costs to be equal? Assume that salvage values for both
motors are negligible and that electricity costs $0.05 per kilowatthour.
Formulation
• Decision Criterion: Minimize Equivalent Uniform
Annual Cost (AC)
AC = CR + Operating cost + Taxes & Ins. + Maintenance
Alpha
Beta
Purchase Price
$12,500
$16,000
Maintenance Cost/yr
500
250
Annual Taxes & Insurance
12,500(0.015)
16,000(0.015)
Efficiency
74%
92%
Useful Life (yrs)
10
10
Note: Electrical Efficiency = power output , and 1 hp = 0.746 kW
• At breakeven, ACa = ACb
$2,490 + $5.04X + $500 + $187 = $3,190 + $4.05X + $250 + $240
or $3,177 + $5.04X = $3,680 + $4.05X. Solving for X, we find X = 508 hrs/year
Sensitivity Graph (Spiderplot)
• Makes explicit the impact of uncertainty in the
estimates of each factor of concern on the
economic measure of merit
–
–
–
–
–
Annual revenue and expenses
Rate of return
Market (or salvage) value
Equipment Life
Capacity utilization
Example
• A machine for which most likely cash flow estimates are
given in the following list is being considered for
immediate installation. Because of the new technology
built into this machine, it is desired to investigate its PW
over a range of 40% in:
• (a) initial investment, (b) annual net cash flow, (c)
salvage value, and (d) useful life
• Based on these estimates, how much can the initial
investment increase without making the machine an
unattractive venture?
• Draw a diagram that summarizes the sensitivity of
present worth to changes in each separate parameter
when the MARR = 10% per year
Example: Solution
PW(10%) = -11,500 + 3,000(P|A,10%,6) +
1,000(P|F,10%,6)= $2,130
a) When the Initial Investment varies by ±p%
PW=(1± p% /100)(-11,500) + 3,000(P|A,10%,6)+ 1,000(P|F,10%,6)
b) When Net Annual Cash Flow varies by ±a%
PW = -11,500+(1± a% /100)(3,000)(P|A, 10%,6)+1,000(P|F,10%,6)
c) When Salvage Value varies by ±s%
PW = -11,500+3,000(P|A,10%,6)+(1± s% /100 )(1,000)(P|F,10%,6)
d) When the Useful Life varies by ±n%
PW = -11,500+3,000[P|A,10%,6(1± n% /100 )]
+1,000[P|F,10%,6(1± n% /100)]
Sensitivity Graph (Spiderplot) Of Four
Factors
PW (10%)
7000
6000
5000
$2130
4000
3000
2000
-% Deviation
Changes in
Factor
Estimate
1000
- 40
-30 -20 -10
-1000
-2000
-3000
-4000
0
+10
+20
+30
+40
+%Deviation
Changes in
Factor
Estimate
Revelations Of Spiderplot
• Shows the sensitivity of the present worth to percent
deviation changes in each factor’s best estimate
• Other factors are assumed to remain at their best estimate
values
• The relative degree of sensitivity of the present worth to
each factor is indicated by the slope of the curves (the
“steeper” the slope of a curve the more sensitive the
present worth is to the factor)
• In this example:
– Present worth is insensitive to MV
– Present worth is sensitive to I, A, and N
Measuring Sensitivity By A
Combination Of Factors
1. Develop a sensitivity graph for the project
2. Use sensitivity graph to select most sensitive project
factors.
3. Analyze combined effects of these factors on project’s
economic measure of merit
Example - Continued
• Which parameter is most sensitive to change?
– PW(10%) = 0 when Initial Investment increases 18.5%
– PW(10%) = 0 when Net Annual CF decreases 16.3%
– PW(10%) = 0 when Salvage Value decreases 378%
• Note: requires a negative Salvage Value
– PW(10%) = 0 when Useful Life decreases 21.7%
Optimistic - Pessimistic Estimates
• Exploring sensitivity by estimating one or more factors
in a favorable direction and in an unfavorable direction
to investigate the effect on study results.
– Optimistic - 95th percentile (desirable)
– Pessimistic - 5th percentile (undesirable)
• Not only do we examine the Equivalent Worth (EW)
for all three estimation conditions, we examine the EW
for all combinations of estimated outcomes for the key
factors being estimated.
Problem 10-18 (page 455)
• Suppose for an engineering project the optimistic, most
likely, and pessimistic estimates are as shown below
Optimistic
Most Likely Pessimistic
Capital Investment -$80,000
-$95,000
-$120,000
Useful Life
12 years
10 years
6 years
Market Value
$30,000
$20,000
0
Net Annual CF
$35,000
$30,000
$20,000
MARR
12%/yr
12%/yr
12%/yr
a) What is the AW for each of the three estimation conditions?
b) It is thought the most critical elements are useful life and net
annual cash flow. Develop a table showing the AW for all
combinations of estimates for these two factors, assuming that all
other factors remain at their most likely values.
Problem 10-18 Set up
• AW(12%) =
• Set up a table to illustrate summary results:
Useful Life
O
ML
P
Net Annual
Cash Flow
$35,000
$30,000
$20,000
O
(12 yrs)
ML
(10 yrs)
P
(6 yrs)
Dealing with Uncertainty
• Uncertainty causes factors to become random variable
in engineering economy analysis
• Risk-Adjusted MARR
– A widely used industrial practice for including some
consideration of uncertainty is to increase the MARR
• Reduction of useful life
– By dropping from consideration those revenues (savings) and
expenses that may occur after a reduced study period, heavy
emphasis is placed on rapid recovery of capital in early years
of a project’s life
– This method is closely related to the discounted payback
technique and suffers from most of the same deficiencies