SMU
EMIS 5300/7300
Systems Analysis Methods
Decision Analysis:
Decision-Making Under Risk
updated 2 December 2005
1
Decision Table
State of the Economy
Stagnant
Investment
Decision
Alternative
Slow
Growth
Rapid
Growth
Maximum
Stocks
-$500
$700
$2200
$2200
Bonds
-$100
$600
$900
$900
CD’s
$300
$500
$750
$750
Mixture
-$200
$650
$1300
$1300
2
Equally Likely (Laplace)
Criterion
• Assumes all states of nature are equally
likely
• Selects the alternative with the maximum
average payoff
Stagnant
Stocks
-500
Bonds
-100
CD's
300
Mixture
-200
Slow Rapid
700 2200
600 900
500 750
650 1300
Average
$800.00
$466.67
$516.67
$583.33
•Selects Stocks in our example
3
Decision-Making Under Risk
• States of nature are not equally likely, but
will occur with know probabilities
– The payoff for each alternative is a random
variable
– Decision-making criteria consider the expected
payoff of each alternative
4
Investment Example
• Suppose that P(stagnant economy) = 0.5,
P(slow growth) = 0.3, and P(rapid growth) = 0.2
• In this case, the payoff for each alternative, stocks,
bonds, cd’s, or mixture, is a random variable.
• One decision criterion is to pick the alternative
that gives the maximum expected monetary value
(EMV).
5
Probability Mass Function of
Payoff from Stocks
• Let X be the payoff from investing in stocks
• The probability mass function of X is
Stagnant
Slow
Rapid
X
P(x)
-500 0.5
700 0.30
2200 0.20
• Expected value of X
  E ( X )   xp( x ) 
x
 500  0.5  700  0.3  2200  0.2  400
6
Expected Monetary Value
Criterion
• Selects the alternative with the maximum
expected (mean) monetary value (payoff)
Stagnant
Prob.
0.5
Stocks
-500
Bonds
-100
CD's
300
Mixture
-200
Slow Rapid
0.30 0.20 EMV
700 2200
$400.00
600 900
$310.00
500 750
$450.00
650 1300
$355.00
CD’s give the maximum EMV.
7
The Expected Value with Perfect
Information
•
Suppose that before we invest, we can consult an
oracle who knows with certainty which state of
nature will occur. Our investment policy is:
1. If the oracle says that the economy will be stagnant,
invest in CD’s and receive a payoff of 300, else
2. if the oracle says that economy will grow slowly,
invest in stocks and receive a payoff of 700, else
3. if the oracle says that economy will grow rapidly,
invest in stocks and receive a payoff of 2,200.
8
The Expected Value with Perfect
Information
•
The outcome of this experiment is also a random
variable,
X
P(x)
300.00
0.5 Stagnant
700.00
0.3 Slow
2,200.00
0.2 Rapid
(X)
800.00
•
The expected value of this random variable is
called the expected value with perfect
information (EVwPI).
9
The Expected Value of Perfect
Information
•
The advantage gained by perfect
information, EVwPI – Max EMV, is
known as the expected value of perfect
information (EVPI).
–
In this case EVPI = $800 – $450 = $350.
10
The Expected Opportunity Loss
Criteria
• An alternative to maximizing the expected
payoff is to minimize the expected
opportunity loss (EOL). That is, minimize
the expected regret.
11
Opportunity Loss Table
State of the Economy
Stagnant
Investment
Decision
Alternative
Slow
Growth
Rapid
Growth
$0
Stocks
$800
$0
Bonds
$400
$100
$1300
$0
$200
$1450
$500
$50
$900
CD’s
Mixture
12
Expected Opportunity Loss
(EOL)
Stagnant Slow Rapid
Regret
Prob.
0.5
0.30
0.20 E(X)
Stocks
800.00
0.00
0.00 400.00
Bonds
400.00 100.00 1,300.00 490.00
CD's
0.00 200.00 1,450.00 350.00
Mixture 500.00 50.00 900.00 445.00
CD’s minimize EOL.
13
Sensitivity Analysis Example
Stagnant
Prob.
0.7
Stocks
-500
Bonds
-100
CD's
300
Mixture
-200
Slow
0.3 EMV
700
-$140.00
600
$110.00
500
$360.00
650
$55.00
CD’s give the maximum EMV.
Let p be the probability of slow growth.
How does our decision change as a function of p?
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Sensitivity Analysis Example
Stagnant
Prob.
0.7
Stocks
-500
Bonds
-100
CD's
300
Mixture
-200
Slow
0.3 EMV
700
-$140.00
600
$110.00
500
$360.00
650
$55.00
EMV (Stocks) = $700 p + (- $500)(1- p) = $1,200 p - $500
EMV(Bonds) = $600 p + (-$100)(1- p) = $700 p - $100
EMV(CD’s) = $500 p + $300(1- p) = $200 p + $300
EMV(Mixture) = $650 p + (-$200)(1- p) = $850 p - $200
15
EMV as a Function of p
P
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
Stocks
-500.00
-380.00
-260.00
-140.00
-20.00
100.00
220.00
340.00
460.00
580.00
700.00
EMV
Bonds CD's
-100.00 300.00
-30.00 320.00
40.00 340.00
110.00 360.00
180.00 380.00
250.00 400.00
320.00 420.00
390.00 440.00
460.00 460.00
530.00 480.00
600.00 500.00
Mixture Row Max
-200.00
300.00
-115.00
320.00
-30.00
340.00
55.00
360.00
140.00
380.00
225.00
400.00
310.00
420.00
395.00
440.00
480.00
480.00
565.00
580.00
650.00
700.00
Switch from CD’s to Mixture for some P in (0.7,0.8).
Switch from Mixture to Stocks for some P in (0.8,0.9).
16
Sensitivity Analysis Example
For which value of P are we indifferent between CD’s and Mixture?
EMV(CD’s) = $500 p + $300(1- p) = $200 p + $300
EMV(Mixture) = $650 p + (-$200)(1- p) = $850 p - $200
$200 p + $300 = $850 p - $200 => $500 = $650 p => p = 0.78.
For which value of P are we indifferent between Stocks and Mixture?
EMV (Stocks) = $700 p + (- $500)(1- p) = $1,200 p - $500
$1,200 p - $500 = $850 p - $200 => $350 p = $300 => p = 0.86
17