Transcript Basic Business Statistics, 10/e

Basic Business Statistics
11th Edition
Chapter 17
Decision Making
Basic Business Statistics, 11e © 2009 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
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-2
Steps in Decision Making

List Alternative Courses of Action


List Uncertain Events


Possible events or outcomes
Determine ‘Payoffs’


Choices or actions
Associate a Payoff with Each Choice/Event
combination
Adopt Decision Criteria

Evaluate Criteria for Selecting the Best Course of
Action
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-3
List Possible Actions or Events
Two Methods
of Listing
Payoff Table
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Decision Tree
Chap 17-4
A Payoff Table
A payoff table shows alternatives,
states of nature, and payoffs
States of Nature
(Events)
Profit in $1,000’s
Investment Choice (Action)
Large
Average
Small
Factory
Factory
Factory
Strong Economy
Stable Economy
Weak Economy
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
200
50
-120
90
120
-30
40
30
20
Chap 17-5
Sample 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
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Chap 17-6
Opportunity Loss
Opportunity loss is the difference between an actual
payoff for an action and the highest possible payoff,
given a particular event
State Of Nature
(Events)
Strong Economy
Stable Economy
Weak Economy
Payoff
Table
Profit in $1,000’s
(Action)
Large
Factory
Average
Factory
Small
Factory
200
50
-120
90
120
-30
40
30
20
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.
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-7
Opportunity Loss
(continued)
States of Nature
(Events)
Strong Economy
Stable Economy
Weak Economy
Profit in $1,000’s
Investment Choice (Action)
Large
Factory
Average
Factory
Small
Factory
200
50
-120
90
120
-30
40
30
20
States of Nature
(Events)
Strong Economy
Stable Economy
Weak Economy
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Payoff
Table
Opportunity
Loss Table
Opportunity Loss in $1,000’s
Investment Choice (Action)
Large
Factory
Average
Factory
Small
Factory
0
70
140
110
0
50
160
90
0
Chap 17-8
Decision Criteria

Maximax


Maximin


The expected opportunity loss for taking action j
Expected Value of Perfect Information (EVPI)


The expected profit for taking action j
Expected Opportunity Loss (EOL(j))


A pessimistic decision criteria
Expected Monetary Value (EMV(j))


An optimistic decision criteria
The expected opportunity loss from the best decision
Return to Risk Ratio

Takes into account the variability in payoffs
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-9
Maximax Solution
States of Nature
(Events)
Strong Economy
Stable Economy
Weak Economy
Profit in $1,000’s
Investment Choice (Action)
Large
Average
Small
Factory
Factory
Factory
200
50
-120
90
120
-30
40
30
20
Maximum payoff for Large Factory is 200
Maximum payoff for Average Factory is 120
Maximum payoff for Small Factory is 40
Maximax Decision: Build the Large Factory because 200 is the maximum
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-10
Maximin Solution
States of Nature
(Events)
Strong Economy
Stable Economy
Weak Economy
Profit in $1,000’s
Investment Choice (Action)
Large
Average
Small
Factory
Factory
Factory
200
50
-120
90
120
-30
40
30
20
Minimum payoff for Large Factory is -120
Minimum payoff for Average Factory is -30
Minimum payoff for Small Factory is 20
Maximin Decision: Build the Small Factory because 20 is the maximum
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-11
Expected Monetary Value
Solution
Goal: Maximize expected value

The expected monetary value is the weighted
average payoff, given specified probabilities for
each event
N
EMV ( j)   x ijPi
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
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-12
Expected Monetary Value
Solution
(continued)

The expected value is the weighted average
payoff, given specified probabilities for each event
Profit in $1,000’s
Investment Choice (Action)
States of Nature
(Events)
Large
Factory
Average
Factory
Small
Factory
Strong Economy (.3)
Stable Economy (.5)
Weak Economy (.2)
200
50
-120
90
120
-30
40
30
20
Suppose these probabilities have been
assessed for these three events
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-13
Expected Monetary Value
Solution
(continued)
Goal: Maximize expected value
Payoff Table:
Profit in $1,000’s
Investment Choice (Action)
States of Nature
(Events)
Large
Factory
Average
Factory
Small
Factory
Strong Economy (.3)
Stable Economy (.5)
Weak Economy (.2)
200
50
-120
90
120
-30
40
30
20
61
81
31
Expected Value (EMV)
Maximize expected value by
choosing Average factory
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Example:
EMV (Average factory) =
(90)(.3) +
(120)(.5) +
(-30)(.2) = 81
Chap 17-14
Decision Trees Are Another Way To
Display & Analyze The Same Information

A Decision tree shows a decision problem,
beginning with the initial decision and ending
will all possible outcomes and payoffs.
Use a square to denote decision nodes
Use a circle to denote uncertain events
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-15
Add Probabilities and Payoffs
(continued)
Large factory
Strong Economy (.3)
200
Stable Economy (.5)
50
Weak Economy
Average factory
Small factory
-120
Strong Economy (.3)
90
Stable Economy (.5)
120
Weak Economy
Decision
(.2)
(.2)
-30
Strong Economy (.3)
40
Stable Economy (.5)
30
Weak Economy
(.2)
20
Uncertain Events
Probabilities Payoffs
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-16
Fold Back the Tree
EMV=200(.3)+50(.5)+(-120)(.2)=61
Large factory
Strong Economy (.3)
200
Stable Economy (.5)
50
Weak Economy
EMV=90(.3)+120(.5)+(-30)(.2)=81
Average factory
Small factory
90
Stable Economy (.5)
120
(.2)
-30
Strong Economy (.3)
40
Stable Economy (.5)
30
Weak Economy
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
-120
Strong Economy (.3)
Weak Economy
EMV=40(.3)+30(.5)+20(.2)=31
(.2)
(.2)
20
Chap 17-17
Make the Decision
EV=61
Large factory
Strong Economy (.3)
200
Stable Economy (.5)
50
Weak Economy
EV=81
Average factory
Strong Economy (.3)
Stable Economy (.5)
Weak Economy
EV=31
Small factory
(.2)
-120
90
Maximum
120
40
Stable Economy (.5)
30
(.2)
EMV=81
-30
Strong Economy (.3)
Weak Economy
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
(.2)
20
Chap 17-18
Expected Opportunity Loss
Solution
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 opportunity loss of action j
Lij = opportunity loss for action j when event i occurs
Pi = probability of event i
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-19
Expected Opportunity Loss
Solution
Goal: Minimize expected opportunity loss
Opportunity Loss Table
Opportunity Loss in $1,000’s
Investment Choice (Action)
States of Nature
(Events)
Large
Factory
Average
Factory
Small
Factory
Strong Economy (.3)
Stable Economy (.5)
Weak Economy (.2)
0
70
140
110
0
50
160
90
0
Expected Op. Loss (EOL)
63
43
93
Minimize expected op. loss by
choosing Average factory
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Example:
EOL (Large factory) =
0(.3) +
70(.5) +
(140)(.2) = 63
Chap 17-20
Expected Opportunity Loss
vs. Expected Monetary Value

The Expected Monetary Value (EMV) and the
Expected Opportunity Loss (EOL) criteria are
equivalent.

Note that in this example the expected monetary
value solution and the expected opportunity loss
solution both led to the choice of the average size
factory.
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-21
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)
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-22
Expected Profit Under Certainty

Expected profit under certainty
= expected value of the best decision, given perfect information
Profit in $1,000’s
Investment Choice (Action)
States of Nature
(Events)
Large
Factory
Average
Factory
Small
Factory
Strong Economy (.3)
Stable Economy (.5)
Weak Economy (.2)
200
50
-120
90
120
-30
40
30
20
Value of best decision
for each event:
200
120
20
Example: Best
decision given
“Strong Economy”
is “Large factory”
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-23
Expected Profit Under Certainty
(continued)
Profit in $1,000’s
Investment Choice (Action)
States of Nature
(Events)
Large
Factory
Average
Factory
Small
Factory
Strong Economy (.3)
Stable Economy (.5)
Weak Economy (.2)
200
50
-120
90
120
-30
40
30
20
Now weight these
outcomes with
their probabilities
to find the
expected value.
200
120
20
200(.3)+120(.5)+20(.2) = 124
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Expected
profit under
certainty
Chap 17-24
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)
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-25
Accounting for Variability
Consider the choice of Stock A vs. Stock B
Percent Return
Stock Choice (Action)
States of Nature
(Events)
Stock A
Stock B
Strong Economy (.7)
30
14
Weak Economy (.3)
-10
8
Expected Return: 18.0
12.2
Stock A has a higher
EMV, but what about
risk?
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Chap 17-26
Accounting for Variability
(continued)
Calculate the variance and standard deviation for
Stock A and Stock B:
States of Nature
(Events)
Percent Return
Stock Choice
(Action)
Stock A
Stock B
Strong Economy (.7)
30
14
Weak Economy (.3)
-10
8
Expected Return:
18.0
Variance:
336.0
Standard Deviation 18.33
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Example
N
σ   ( X i  μ)2 P( X i ) 
2
A
i 1
(30  18) 2 (.7)  (10  18) 2 (.3)  336.0
12.2
7.56
2.75
Chap 17-27
Accounting for Variability
(continued)
Calculate the coefficient of variation for each stock:
CVA 
σA
18.33
 100% 
 100%  101.83%
EMV A
18.0
CVB 
σB
2.75
 100% 
 100%  22.54%
EMV B
12.2
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Stock A has
much more
relative
variability
Chap 17-28
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)
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-29
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
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-30
Decision Making
with Sample Information
Prior
Probability

Permits revising old
probabilities based on new
information
New
Information
Revised
Probability
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Chap 17-31
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
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
P(F1 | E2) = 0.30
Chap 17-32
Revised Probabilities
Example
(continued)
P(F1 | E1)  .9 , P(F1 | E2 )  .3
P(E1 )  .7 , P(E2 )  .3

Revised Probabilities (Bayes’ Theorem)
P(E1 )P(F1 | E1 )
(.7)(.9)
P(E1 | F1 ) 

 .875
P(F1 )
(.7)(.9)  (.3)(.3)
P(E2 )P(F1 | E2 )
P(E2 | F1 ) 
 .125
P(F1 )
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-33
EMV with
Revised Probabilities
Pi
Event
Stock A
xijPi
Stock B
xijPi
.875
strong
30
26.25
14
12.25
.125
weak
-10
-1.25
8
1.00
Σ = 25.0
Revised
probabilities
Σ = 13.25
EMV Stock B = 13.25
EMV Stock A = 25.0
Maximum
EMV
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Chap 17-34
EOL Table with
Revised Probabilities
Pi
Event
Stock A
xijPi
Stock B
xijPi
.875
strong
0
0
16
14.00
.125
weak
18
2.25
0
0
Σ = 2.25
Revised
probabilities
Σ = 14.00
EOL Stock B = 14.00
EOL Stock A = 2.25
Minimum
EOL
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-35
Accounting for Variability with
Revised Probabilities
Calculate the variance and standard deviation for
Stock A and Stock B:
Percent Return
Stock Choice (Action)
States of Nature
(Events)
Stock A
Example
Stock B
N
σ   (X i  μ)2 P(X i ) 
2
A
i 1
Strong Economy (.875)
30
14
Weak Economy (.125)
-10
8
Expected Return:
25.00
Variance:
175.00
Standard Deviation: 13.23
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
(30  25) 2 (.875)  (10  25) 2 (.125)  175.0
13.25
3.94
1.98
Chap 17-36
Accounting for Variability with
Revised Probabilities
(continued)
The coefficient of variation for each stock using the
results from the revised probabilities:
CVA 
σA
13.229
 100% 
 100%  52.92%
EMV A
25.0
CVB 
σB
1.984
 100% 
 100%  14.97%
EMV B
13.25
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-37
Return-to-Risk Ratio with
Revised Probabilities
EMV(A)
25.0
RTRR(A) 

 1.890
σA
13.229
EMV(B) 13.25
RT RR(B) 

 6.677
σB
1.984
With the revised probabilities, both stocks have
higher expected returns, lower CV’s, and larger
return to risk ratios
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-38
Utility

Utility is the pleasure or satisfaction
obtained from an action.

The utility of an outcome may not be the same
for each individual.
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-39
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.
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-40
Three Types of Utility Curves
$
Risk Averter
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
$
Risk Seeker
$
Risk-Neutral
Chap 17-41
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
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-42
Chapter Summary

Described the payoff table and decision trees


Provided criteria for decision making







Opportunity loss
Maximax and Maximin
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
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 17-43