1. Systems Analysis Methods
SMU
EMIS 5300/7300
Systems Analysis Methods
Decision Analysis:
Decision-Making Under Risk
updated 2 December 2005

2. Decision Table State of the Economy Stagnant Slow Rapid
Growth Growth Maximum
Stocks -$ $700 $ $2200
Investment Bonds -$ $600 $ $900
Decision
Alternative CD’s $ $500 $ $750
Mixture -$ $650 $ $1300

3. Equally Likely (Laplace) Criterion
Assumes all states of nature are equally likely
Selects the alternative with the maximum average payoff
Selects Stocks in our example

4. 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

5. 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).

6. Probability Mass Function of Payoff from Stocks
Let X be the payoff from investing in stocks
The probability mass function of X is
Expected value of X

7. Expected Monetary Value Criterion
Selects the alternative with the maximum expected (mean) monetary value (payoff)
CD’s give the maximum EMV.

8. 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:
If the oracle says that the economy will be stagnant, invest in CD’s and receive a payoff of 300, else
if the oracle says that economy will grow slowly, invest in stocks and receive a payoff of 700, else
if the oracle says that economy will grow rapidly, invest in stocks and receive a payoff of 2,200.

9. The Expected Value with Perfect Information
The outcome of this experiment is also a random variable,
The expected value of this random variable is called the expected value with perfect information (EVwPI).

10. 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.

11. 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.

12. Opportunity Loss Table
State of the Economy
Stagnant Slow Rapid
Growth Growth
Stocks $ $ $0
Investment Bonds $400 $100 $1300
Decision
Alternative CD’s $0 $200 $1450
Mixture $ $50 $900

13. Expected Opportunity Loss (EOL)
CD’s minimize EOL.

14. Sensitivity Analysis Example
CD’s give the maximum EMV.
Let p be the probability of slow growth.
How does our decision change as a function of p?

15. Sensitivity Analysis Example
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

16. EMV as a Function of p
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).

17. 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