Dealing With Uncertainty

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CHAPTER 10

• Dealing With Uncertainty

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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

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Four Major Sources Of Uncertainty

• Possible inaccuracy of cash-flow estimates used
  in the study
• How much source information is available
• How dependable is the source information
• 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
• Type of physical plant and equipment involved
• Some equipment have more definite lives and MV
  than the others (general lathe machine vs. mining
  equipment)
• Length of study period
• The longer the study period, the greater the
  level of uncertainty of a capital investment

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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
• Breakeven Analysis
• Sensitivity Graph (spiderplot)
• Combination of factors

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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.

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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

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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 kilowatt-hour.

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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

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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

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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

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Example Solution

• PW(10) -11,500 3,000(PA,10,6)
  1,000(PF,10,6) 2,130
• a) When the Initial Investment varies by p
• PW(1 p /100)(-11,500) 3,000(PA,10,6)
  1,000(PF,10,6)
• b) When Net Annual Cash Flow varies by a
• PW -11,500(1 a /100)(3,000)(PA,
  10,6)1,000(PF,10,6)
• c) When Salvage Value varies by s
• PW -11,5003,000(PA,10,6)(1 s /100
  )(1,000)(PF,10,6)
• d) When the Useful Life varies by n
• PW -11,5003,000PA,10,6(1 n /100 )
• 1,000PF,10,6(1 n /100)

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Sensitivity Graph (Spiderplot) Of Four Factors
PW (10)
7000
6000
Capital Investment
5000
Annual Net Cash Flow, A
2130
Useful Life, N
4000
3000
Market Value, MV
2000

• Deviation Changes in Factor Estimate

• Deviation
• Changes in
• Factor
• Estimate

1000
0
10 20 30 40
- 40 -30 -20 -10
-1000
-2000
-3000
-4000
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Revelations Of Spiderplot

• Shows the sensitivity of the present worth to
  percent deviation changes in each factors 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

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Measuring Sensitivity By A Combination Of Factors

• Develop a sensitivity graph for the project
• Use sensitivity graph to select most sensitive
  project factors.
• Analyze combined effects of these factors on
  projects economic measure of merit

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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

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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.

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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.

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Problem 10-18 Set up

• AW(12)
• Set up a table to illustrate summary results
• Useful Life
• Net Annual O ML P
• Cash Flow (12 yrs) (10 yrs) (6 yrs)
• O 35,000
• ML 30,000
• P 20,000

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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
  projects life
• This method is closely related to the discounted
  payback technique and suffers from most of the
  same deficiencies