1. Dealing With Uncertainty
CHAPTER 10
Dealing With Uncertainty

2. 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

3. 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

4. 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

5. 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.

6. 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

7. 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.

8. Formulation
Decision Criterion: Minimize Equivalent Uniform Annual Cost (AC)
AC = CR + Operating cost + Taxes & Ins. + Maintenance
Alpha Beta
Purchase Price $12, $16,000
Maintenance Cost/yr
Annual Taxes & Insurance 12,500(0.015) 16,000(0.015)
Efficiency % 92%
Useful Life (yrs)
Note: Electrical Efficiency = power output , and 1 hp = 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

9. 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

10. 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

11. Example: Solution
PW(10%) = -11, ,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)]

12. Sensitivity Graph (Spiderplot) Of Four Factors
PW (10%)
7000
Capital Investment
6000
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
-1000
-2000
-3000
-4000

13. 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

14. 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 project’s economic measure of merit

15. 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%

16. 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.

17. Problem (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.

18. Problem 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

19. 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