Dealing%20with%20Uncertainty

1
Dealing with Uncertainty

• Probabilistic Risk Analysis

2
Introduction

• Statistical and probability concepts can also be
  used to analyze the economic consequences of some
  decision situations involving risk and
  uncertainty.
• The probability that a cost, revenue, useful
  life, NPW, etc. will occur is usually considered
  to be the long-run relative frequency with which
  the value occurs.

3
Random Variables

• Factors having probabilistic outcomes are called
  random variables.
• Useful information about a random variable is
• expected value (average, mean), denoted by EX
• variance, denoted by VarX
• standard deviation, denoted by SDX
• When uncertainty is considered, the
• variability in the economic measures of merit and
  
• the probability of loss associated with the
  alternative are very useful in the
  decision-making process.

4
Some Important Relationships

• Discrete Random Variables
• Probability PrXxi p(xi) for i 1, ..., L
• where p(xi)gt0 and ?i p(xi)1
• Continuous Random Variables
• Probability
  where
• Expected Value
• EX ?i xi p(xi) or
• Variance
• VarX ?i (xi - EX) 2p(xi) or

5
Some Important Relationships (contd)

• Variance VarX EX2 - (EX)2
• Standard Deviation SDX (VarX) 1/2
• Expected value of a sum EXY EX EY
• Variance of a sum or differenceVarXY
  VarX-Y VarX VarY
• when X and Y are independent
• Multiply by a constant EcX cEX and VarcX
  c2 VarX
• Expected Value of a function Eh(X) ?i h(xi)
  p(xi) or

6
Evaluation of Projects with Random Outcomes

• We can use the expected value and variance
  concepts to assess the projects worth
• We might be interested in
• the expected net present worth, ENPW, or
• expected net annual worth, ENAW
• the variance or standard deviation of the
  traditional measures, VarNPW, VarNAW,
  SDNPW, SDNAW
• the probability that the NPW or NAW is positive,
  i.e., ProbNPW gt 0 or ProbNAWgt0

7
Example 6

• Three alternatives are being evaluated for the
  protection of electrical circuits in a large
  manufacturing plant. Annual costs are 10 of
  capital investments.

• If a loss of power occurs, it will cost 80,000
  with a probability of 0.65 and 120,000 with a
  probability of 0.35. Useful life is 8 years. MARR
  12.

8
Example 6 (contd)

• Expected cost of a loss
• -80,000 (0.65) -120,000 (0.35) -94,000

• Best based on total expected NAW Alternative B

9
Example 7

• A HVAC system has become unreliable and
  inefficient. Rental income is being hurt and OM
  continue to increase. You decide to rebuild it.
• Assume MARR 12

10
Example 7 (contd)

• For year 12,
• NPW -521,000 (48,60031,000) (P/A, 12,
  12)
• -27,926
• However, this useful life only has a 0.1 chance
  of occurring.
• For year 13,
• NPW -521,000 (48,60031,000) (P/A, 12,
  13)
• -9,689
• However, this useful life only has a 0.2 chance
  of occurring.

11
Example 7 (contd)

• What is ENPW and VarNPW ?

12
Example 7 (contd)

• ENPW 9,984
• E(NPW)2 (2) 577.524 x 106
• VarNPW E(NPW)2 - (ENPW)2
• (2)477.847 x 106
• SDNPW (VarNPW)1/2 21,859
• ProbabilityNPW gt 0 1- (0.10.2) 0.7
• The weakest indicator is SD(NPW) gt 2ENPW !

13
Example 8

• For the following cash flow estimates, find
  ENPW, VarNPW, and SDNPW. Determine Prob
  ROR lt MARR. Assume that the annual net cash
  flows are normally distributed and independent.
  Use a MARR 15.

14
Example 8 (contd)

• The investment is known.
• Year 0

15
Example 8 (contd)

• The cash flows for the years 1, 2 and 3 are not
  known.

16
Example 8 (contd)

• ENPW -7,000 3,500 (P/F,15,1)
  3,000 (P/F,15,2) 2,800 (P/F,15,3)
• 153
• VarNPW 02 (600)2 (P/F,15,1)2
  (500)2 (P/F,15,2)2 (400 )2 (P/F,15,3)2
• (2 )484,324
• SDNPW 696

17
Example 8 (contd)

• Prob ROR lt MARR ?
• Step 1 For a project having a unique ROR (simple
  investments are such projects), the probability
  that the ROR is less than the MARR is the same as
  the probability that the NPW is less than 0. So
• Prob ROR lt MARR Prob NPW lt 0
• Step 2 Because the NPW is normally distributed,
  we can normalize to a N(0,1) distribution. So
• Z (NPW - ENPW)/SD(NPW) (0-153)/696 -0.22
• Step 3 Using Normal Tables, we get
• ProbNPW lt0 ProbZ lt -0.22 0.4129
• Therefore Prob ROR lt MARR 0.4129

18
Decision Trees

• Also called decision flow networks and decision
  diagrams
• Powerful means of depicting and facilitating the
  analysis of problems involving sequential
  decisions and variable outcomes over time
• Make it possible to break down large, complicated
  problems into a series of smaller problems

19
Diagramming

• Square symbol depicts a decision node
• Circle symbol depicts a chance outcome node
• All initial or immediate alternatives among which
  the decision maker wishes to choose
• All uncertain outcomes and future alternatives
  that may directly affect the consequences
• All uncertain outcomes that may provide
  information

20
Diagramming Example

Sales
good
bad
Invest in new product line
Decision
Status Quo
21
Example 9

• A new design is being evaluated as potential
  replacement for a heavily used machine. The new
  design involves major changes that have expected
  advantage, but would be 8600 more expensive. In
  return, annual expense savings are expected, but
  their extent depend on the machines reliability.

• Use MARR 18. Life 6 years. Salvage 0.

22
Example 9 (contd)
A 3,470
NPW 3,538
25
A 2,920
NPW 1,614
40
25
A 2,310
NPW -520
New Design
10
A 1,560
NPW -3,143
Current Design
23
Example 9 (contd)

• Based on a before-tax analysis (MARR 18,
• analysis period 6 years, salvage value 0),
• is the new design economically preferable to the
  current unit?
• ENPW - 8600 0.25 (3470) (P/A,18,6)
• 0.4 (2920) (P/A,18,6)
• 0.25 (2310) (P/A,18,6)
• 0.10 (1560)(P/A,18,6)
• 1086

24
Example 9 (contd)
A 3,470
NPW 3,538
25
1,086
A 2,920
NPW 1,614
40
25
A 2,310
NPW -520
New Design
10
A 1,560
NPW -3,143
Current Design
0
25
Example 9 (contd)

• Optimal decision based on perfect information
• Expected Value of Perfect Information
• 1530 - 1086 444

26
Example 9 (contd)

• Management is confident that data from an
  additional comprehensive test will show whether
  future operational performance will be favorable
  (excellent or good reliability) or not favorable
  (standard or poor reliability). The design team
  develop conditional probability estimates.

27
Example 9 (contd)

• We need to determine the joint probabilities of
  the design goal being met at a particular level
  and a certain test outcome occurring.
• For example,
• p(E, F) p(FE) p(E) (0.95)(0.25) 0.2375
• p(E,NF) p(NFE) p(E) (0.05)(0.25) 0.0125

28
Example 9 (contd)

• The revised probabilities of each outcome are
  obtained from the joint probabilities and the
  marginal probabilities
• For example, when favorable
• p(E) p(E,F)/p(F) 0.2375/0.6575 0.3612
• When not favorable
• p(E) p(E,NF)/p(NF) 0.0125/0.3425 0.0365

29
Example 9 (contd)
1086

E 0.3612
3538
No test
G 0.5171
1614
New Design
S 0.1141
-520
P 0.0076
-3143
Do test
Current Design
Favorable
E 0.0365
3538
G 0.1752
1614
New Design
S 0.5109
-520
Unfavorable
P 0.2774
-3143
Current Design
30
Example 9 (contd)
1086

No test
2029
2029
2029
New Design
0
Do test
Current Design
Favorable
-726
New Design
0
Unfavorable
Current Design
0