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Statistics for Managers
Using Microsoft® Excel
5th Edition
Chapter 17
Decision Making
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-1
Learning Objectives
In this chapter, you learn:
 To use payoff tables and decision trees to
evaluate alternative courses of action
 To use several criteria to select an alternative
course of action
 To use Bayes’ theorem to revise probabilities
in light of sample information
 About the concept of utility
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-2
Steps in Decision Making
 List Alternative Courses of Action
 Choices or actions
 List Uncertain Events
 Possible events or outcomes
 Determine ‘Payoffs’
 Associate a Payoff with Each Event/Outcome
combination
 Adopt Decision Criteria
 Evaluate Criteria for Selecting the Best Course
of Action
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-3
Payoff Table
A payoff table shows alternatives, states
of nature, and payoffs
Profit in $1,000’s
(Events)
Strong Economy
Stable Economy
Weak Economy
Investment Choice
(Action)
Large
Average
Small
Factory
Factory
Factory
200
50
-120
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
90
120
-30
40
30
20
Chap 17-4
Decision Tree
Large factory
Average factory
Small factory
Strong Economy
200
Stable Economy
50
Weak Economy
-120
Strong Economy
90
Stable Economy
120
Weak Economy
-30
Strong Economy
40
Stable Economy
30
Weak Economy
20
Payoffs
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-5
Opportunity Loss
Opportunity loss is the difference between an actual payoff
for an action and the optimal payoff, given a particular event
Profit in $1,000’s
(Events)
Strong Economy
Stable Economy
Weak Economy
Investment Choice
(Action)
Large
Factory
Average
Factory
Small
Factory
200
50
-120
90
120
-30
40
30
20
Payoff
Table
The action “Average factory” has payoff 90 for “Strong Economy”. Given “Strong
Economy”, the choice of “Large factory” would have given a payoff of 200, or
110 higher. Opportunity loss = 110 for this cell.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-6
Opportunity Loss
Investment Choice (Action)
Profit in $1,000’s
(Events)
Strong Economy
Stable Economy
Weak Economy
Large
Factory
Average
Factory
Small
Factory
200
50
-120
90
120
-30
40
30
20
Payoff
Table
Opportunity
Loss Table
Investment Choice (Action)
Opportunity Loss in
$1,000’s
(Events)
Strong Economy
Stable Economy
Weak Economy
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Large
Factory
Average
Factory
Small
Factory
0
70
140
110
0
50
160
90
0
Chap 17-7
Decision Criteria
 Expected Monetary Value (EMV)
 The expected profit for taking action Aj
 Expected Opportunity Loss (EOL)
 The expected opportunity loss for taking action Aj
 Expected Value of Perfect Information (EVPI)
 The expected opportunity loss from the best
decision
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-8
Expected Monetary Value
Goal: Maximize expected value
 The expected monetary value is the weighted
average payoff, given specified probabilities for each
event
N
EMV ( j )   X ij Pi
i 1
Where EMV(j) = expected monetary value of action j
Xij = payoff for action j when event i occurs
Pi = probability of event i
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-9
Expected Monetary Value
 The expected value is the weighted average
payoff, given specified probabilities for
each event
Profit in $1,000’s
(Events)
Strong Economy (0.3)
Stable Economy (0.5)
Weak Economy (0.2)
Investment Choice
(Action)
Large
Factory
Average
Factory
Small Factory
200
50
-120
90
120
-30
40
30
20
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Suppose these
probabilities
have been
assessed for
these three
events
Chap 17-10
Expected Monetary Value
Payoff Table:
Profit in $1,000’s
(Events)
Strong Economy (0.3)
Stable Economy (0.5)
Weak Economy (0.2)
EMV (Expected Values)
Investment Choice
(Action)
Large
Factory
Average
Factory
Small
Factory
200
50
-120
90
120
-30
40
30
20
61
81
31
 Example: EMV (Average factory) = 90(.3) + 120(.5)
+ (-30)(.2) = 81
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-11
Expected Opportunity Loss
Goal: Minimize expected opportunity loss
 The expected opportunity loss is the weighted
average loss, given specified probabilities for each
event
N
EOL(j)   LijPi
i 1
Where EOL(j) = expected monetary value of action j
Lij = opportunity loss for action j when event i occurs
Pi = probability of event i
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-12
Expected Opportunity Loss
Opportunity Loss Table
Investment Choice (Action)
Opportunity Loss in
$1,000’s
(Events)
Large
Factory
Average
Factory
Small
Factory
Strong Economy (0.3)
Stable Economy (0.5)
Weak Economy (0.2)
0
70
140
110
0
50
160
90
0
Expected Opportunity
Loss (EOL)
63
43
93
 Example: EOL (Large factory) = 0(.3) + 70(.5) +
(140)(.2) = 63
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-13
Value of Information
Expected Value of Perfect Information, EVPI
 Expected Value of Perfect Information
EVPI = Expected profit under certainty
– expected monetary value of the best
alternative
 EVPI is equal to the expected opportunity loss
from the best decision
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-14
Expected Profit Under Certainty
 Expected
profit under
certainty
= expected
value of the
best decision,
given perfect
information
Investment Choice
(Action)
Profit in $1,000’s
(Events)
Strong Economy (0.3)
Stable Economy (0.5)
Weak Economy (0.2)
Value of best decision
for each event:
Large
Factory
Average
Factory
Small Factory
200
50
-120
90
120
-30
40
30
20
200
120
20
Example: Best decision given “Strong
Economy” is “Large factory”
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-15
Expected Profit Under Certainty
Investment Choice
(Action)
Profit in $1,000’s
(Events)
 Now weight
these outcomes
with their
probabilities to
find the
expected profit
under certainty:
Strong Economy (0.3)
Stable Economy (0.5)
Weak Economy (0.2)
Large
Factory
Average
Factory
Small Factory
200
50
-120
90
120
-30
40
30
20
200
120
20
200(.3)+120(.5)+20(.2)
= 124
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-16
Value of Information Solution
 Expected Value of Perfect Information (EVPI)
EVPI = Expected profit under certainty
– Expected monetary value of the best decision
Recall: Expected profit under certainty = 124
EMV is maximized by choosing “Average factory,”
where EMV = 81
so:
EVPI = 124 – 81
= 43
(EVPI is the maximum you would be willing to spend to obtain perfect information)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-17
Accounting for Variability
Consider the choice of Stock A vs. Stock B
Stock Choice
(Action)
Percent Return
(Events)
Stock A
Strong Economy
(.7)
30
14
Weak Economy
(.3)
-10
8
Expected Return
(EMV)
18.0
12.2
Stock B
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Stock A has a higher
EMV, but what about
risk?
Chap 17-18
Accounting for Variability
Calculate the variance and standard deviation
Stock Choice
(Action)
Percent Return
(Events)
Stock A
Strong Economy (.7)
30
14
Weak Economy (.3)
-10
8
Expected Return (EMV)
18.0
12.2
Variance
336.0
7.56
Standard Deviation
18.33
2.75
Example:
Stock B
N
σ   ( Xi  μ)2 P( Xi )  (30  18)2 (.7)  ( 10  18)2 (.3)  336.0
2
A
i1
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-19
Accounting for Variability
Calculate the coefficient of variation for each stock:
CVA 
σA
18.33
 100% 
 100%  101.83%
EMVA
18.0
CVB 
σB
2.75
 100% 
 100%  22.54%
EMVB
12.2
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Stock A has
much more
relative
variability
Chap 17-20
Return-to-Risk Ratio
Return-to-Risk Ratio (RTRR):
EMV(j)
RTRR(j) 
σj
Expresses the relationship between the return (expected
payoff) and the risk (standard deviation)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-21
Return-to-Risk Ratio
RTRR(j) 
EMV(j)
σj
RTRR(A) 
EMV(A) 18.0

 0.982
σA
18.33
RTRR(B) 
EMV(B) 12.2

 4.436
σB
2.75
You might want to consider Stock B if you don’t like risk.
Although Stock A has a higher Expected Return, Stock B has a
much larger return to risk ratio and a much smaller CV.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-22
Decision Making
with Sample Information
 Permits revising old
probabilities based on new
information
Prior
Probability
New
Information
Revised
Probability
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-23
Revised Probabilities
Example
Additional Information: Economic forecast is strong economy
 When the economy was strong, the forecaster was correct 90% of the time.
 When the economy was weak, the forecaster was correct 70% of the time.
F1 = strong forecast
F2 = weak forecast
E1 = strong economy = 0.70
Prior probabilities
from stock choice
example
E2 = weak economy = 0.30
P(F1 | E1) = 0.90
P(F1 | E2) = 0.30
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-24
Revised Probabilities
Example
P(F1 | E1)  .9 , P(F1 | E2 )  .3
P(E1)  .7 , P(E2 )  .3
Revised Probabilities (Bayes’ Theorem)
P( F1 | E1 ) P( E1 )
(.9)(. 7)
P( E1 | F1 ) 

 .875
P( F1 )
(.9)(. 7)  (.3)(. 3)
P( F1 | E2 ) P( E2 )
P( E2 | F1 ) 
 .125
P( F1 )
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-25
EMV with
Revised Probabilities
Pi
Event
Stock A
XijPi
.875
strong
30
26.25
14
12.25
.125
weak
-10
-1.25
8
1.00
Σ = 25.0
Revised
probabilities
Stock B
XijPi
Σ = 13.25
EMV Stock B = 13.25
EMV Stock A = 25.0
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Maximum
EMV
Chap 17-26
EOL Table with
Revised Probabilities
Pi
Event
Stock A
XijPi
Stock B
.875
strong
0
0
16
14.00
.125
weak
18
2.25
0
0
Σ = 2.25
Revised
probabilities
XijPi
Σ = 14.00
EOL Stock B = 14.00
EOL Stock A = 2.25
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Minimum
EOL
Chap 17-27
Accounting for Variability with
Revised Probabilities
Calculate the variance and standard deviation
Stock Choice
(Action)
Percent Return
(Events)
Stock A
Stock B
Strong Economy (.875)
30
14
Weak Economy (.125)
-10
8
Expected Return (EMV)
25.0
13.25
Variance
175.0
3.94
Standard Deviation
13.229
1.984
Example:
N
σ   ( X i  μ)2 P( X i )  (30  25) 2 (.875)  (10  25) 2 (.125)  175.0
2
A
i 1
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-28
Accounting for Variability with
Revised Probabilities
The coefficient of variation for each stock using the
results from the revised probabilities:
CVA 
σA
13.229
 100% 
 100%  52.92%
EMVA
25.0
CVB 
σB
1.984
 100% 
 100%  14.97%
EMVB
13.25
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-29
Return-to-Risk Ratio with
Revised Probabilities
EMV(A)
25.0
RTRR(A) 

 1.890
σA
13.229
EMV(B) 13.25
RTRR(B) 

 6.678
σB
1.984
With the revised probabilities, both stocks have higher
expected returns, lower CV’s, and larger return to risk
ratios
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-30
Utility
 Utility is the pleasure or satisfaction obtained
from an action.
 The utility of an outcome may not be the
same for each individual.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-31
Utility Example
Each incremental $1 of profit does not have the
same value to every individual:
A risk averse person, once reaching a goal,
assigns less utility to each incremental $1.
A risk seeker assigns more utility to each
incremental $1.
A risk neutral person assigns the same utility
to each extra $1.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-32
Three Types of Utility Curves
$
Risk Averter
$
Risk Seeker
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
$
Risk-Neutral
Chap 17-33
Maximizing Expected Utility
 Making decisions in terms of utility, not $
 Translate $ outcomes into utility outcomes
 Calculate expected utilities for each action
 Choose the action to maximize expected utility
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-34
Chapter Summary
In this chapter, we have
 Described the payoff table and decision trees
 Opportunity loss
 Provided criteria for decision making
 Expected monetary value
 Expected opportunity loss
 Return to risk ratio
 Introduced expected profit under certainty and the
value of perfect information
 Discussed decision making with sample information
 Addressed the concept of utility
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 17-35