1. Module C2 Decision Models Decision Making Under Risk

2. When doing decision making under uncertainty, we assumed we had “no idea” about which state of nature would occur. In decision making under risk, we assume we have some idea (by experience, gut feel, experiments, etc.) about the likelihood of each state of nature occuring

3. The Expected Value Approach Given a set of probabilities for the states of nature, p 1, p 2 … etc., for each decision an expected payoff can be calculated by:  p i (payoff i ) If this is a decision that will be repeated over and over again, the decision with the highest expected payoff should be the one selected to maximize total expected payoff. But if this is a one-time decision, perhaps the risk of losing much money may be too great -- thus the expected payoff is just another piece of information to be considered by the decision maker.

4. Expected Value Decision for Tom Brown Suppose the broker has offered his own projections for the probabilities of the states of nature: P(S 1 ) =.2, P(S 2 ) =.3, P(S 3 ) =.3, P(S 4 ) =.1, P(S 5 ) =.1 -$100 -$150-$100$150$200$250 $60 -$200-$600 $100$200$300$0 S1 Lg Rise S2 Sm Rise S3 No Chg. S4 Sm Fall S5 Lg Fall D1: Gold D2: Bond D3: Stock D4: C/D$60 $500 $60 $250$100.2.3.1 Probability Expected Value.2(-100)+.3(100)+.3(200)+.1(300)+.1(0).2(250)+.3(200)+.3(150) +.1(-100)+.1(-150).2(500)+.3(250)+.3(100) +.1(-200)+.1(-600).2(60)+.3(60)+.3(60) +.1(60)+.1(60) $100 $130 $125 $60 Highest -- Choose D2 - Bond

5. Perfect Information Although the states of nature are assumed to occur with the previous probabilities, suppose you knew, each time which state of nature would occur -- i.e. you had perfect information Then when you knew S 1 was going to occur, you would make the best decision for S 1 (Stock = $500). This would happen p 1 =.2 of the time. When you knew S 2 was going to occur, you would make the best decision for S 2 (Stock = $250). This would happen p 1 =.3 of the time. And so forth

6. Expected Value of Perfect Information (EVPI) The expected value of perfect information (EVPI) is the gain in value from knowing for sure which state of nature will occur when, versus only knowing the probabilities. It is the upper bound on the value of any additional information.

7. Calculating the EVPI -$100 -$150-$100$150$200$250 $60 -$200-$600 $100$200$300$0 S1 Lg Rise S2 Sm Rise S3 No Chg. S4 Sm Fall S5 Lg Fall D1: Gold D2: Bond D3: Stock D4: C/D$60 $500 $60 $250$100.2.3.1 Probability Expected Return With Perfect Information (ERPI) =.2(500) +.3(250) +.3(200) +.1(300) +.1(60) = $271 Expected Return With No Additional Information = EV(Bond) = $130 Expected Value Of Perfect Information (EVPI) = ERPI - EV(Bond) = $271 - $130 = $141

8. Using the Decision Template Enter Probabilities Expected Value Decision EVPI

9. Sample Information One never really has perfect information, but can gather additional information, get expert advice, etc. that can indicate which state of nature is likely to occur each time. The states of nature still occur, in the long run with P(S 1 ) =.2, P(S 2 ) =.3, P(S 3 ) =.3, P(S 4 ) =.1, P(S 5 ) =.1. We need a strategy of what to do given each possibility of the indicator information We want to know the value of this sample information (EVSI).

10. Sample Information Approach Given the outcome of the sample information, we revise the probabilities of the states of nature occurring (using Bayesian analysis). Then we repeat the expected value approach (using these revised probabilities) to see which decision is optimal given each possible value of the sample information.

11. Example -- Samuelman Forecast Noted economist Milton Samuelman gives an economic forecast indicating either Positive or Negative economic growth in the coming year. Using a relative frequency approach based on past data it has been observed: P(Positive|large rise) =.8 P(Negative|large rise) =.2 P(Positive|small rise) =.7 P(Negative|small rise) =.3 P(Positive|no change)=.5 P(Negative|no change)=.5 P(Positive|small fall) =.4 P(Negative|small fall) =.6 P(Positive|large fall) = 0 P(Negative|large fall) = 1

12. Bayesian Probabilities Given a Positive Forecast Prob(Positive) = P(Positive and Large Rise) + P(Positive and Small Rise) + P(Positive and No Change) + P(Positive and Small Fall) + P(Positive and Large Fall) Prob(Positive) = P(Positive|Large Rise)P(Large Rise) + P(Positive|Small Rise) P(Small Rise) + P(Positive|No Change)P(No Change) + P(Positive|Small Fall) P(Small Fall) + P(Positive|Large Fall) P(Large Fall) (.80)(.20) (.70) (.50) (.30) (.40)(.10) (0) (.10) =.56 P(Large Rise|Pos) = P(Pos|Lg. Rise)P(Lg. Rise)/P(Pos) P(Small Rise|Pos) = P(Pos|Sm. Rise)P(Sm. Rise)/P(Pos) P(No Change|Pos) = P(Pos|No Chg.)P(No Chg.)/P(Pos) P(Small Fall|Pos) = P(Pos|Sm. Fall)P(Sm. Fall)/P(Pos) P(Large Fall|Pos) = P(Pos|Lg. Fall)P(Lg. Fall)/P(Pos) (.80) (.20) /.56 =.286 (.70) (.30) /.56 =.375 (.50) (.30) /.56 =.268 (.40) (.10) /.56 =.071 (0) (.10) /.56 = 0

13. Best Decision With Positive Forecast -$100 -$150-$100$150$200$250 $60 -$200-$600 $100$200$300$0 S1 Lg Rise S2 Sm Rise S3 No Chg. S4 Sm Fall S5 Lg Fall D1: Gold D2: Bond D3: Stock D4: C/D$60 $500 $60 $250$100.286.375.268.0710 Revised Probability Expected Value $84 $180 $249 $60 Highest With Positive Forecast -- Choose D3 - Stock When Samuelman predicts “positive” -- Choose the Stock!

14. Bayesian Probabilities Given a Negative Forecast Prob(Negative) = P(Negative and Large Rise) + P(Negative and Small Rise) + P(Negative and No Change) + P(Negative and Small Fall) + P(Negative and Large Fall) Prob(Negative) = P(Negative|Large Rise)P(Large Rise) + P(Negative|Small Rise) P(Small Rise) + P(Negative|No Change)P(No Change) + P(Negative|Small Fall) P(Small Fall) + P(Negative|Large Fall) P(Large Fall) (.20) (.30) (.50) (.30) (.60)(.10) (1) (.10) =.44 P(Large Rise|Neg) = P(Neg|Lg. Rise)P(Lg. Rise)/P(Neg) P(Small Rise|Neg) = P(Neg|Sm. Rise)P(Sm. Rise)/P(Neg) P(No Change|Neg) = P(Neg|No Chg.)P(No Chg.)/P(Neg) P(Small Fall|Neg) = P(Neg|Sm. Fall)P(Sm. Fall)/P(Neg) P(Large Fall|Neg) = P(Neg|Lg. Fall)P(Lg. Fall)/P(Neg) (.20) (.20) /.44 =.091 (.30) (.30) /.44 =.205 (.50) (.30) /.44 =.341 (.60) (.10) /.44 =.136 (1) (.10) /.44 =.227

15. Best Decision With Negative Forecast -$100 -$150-$100$150$200$250 $60 -$200-$600 $100$200$300$0 S1 Lg Rise S2 Sm Rise S3 No Chg. S4 Sm Fall S5 Lg Fall D1: Gold D2: Bond D3: Stock D4: C/D$60 $500 $60 $250$100.091.205.341.136.227 Revised Probability Expected Value $120 $ 67 -$33 $60 Highest With Negative Forecast -- Choose D1 - Gold When Samuelman predicts “negative” -- Choose Gold!

16. Strategy With Sample Information If the Samuelman Report is Positive -- –Choose the stock! If the Samuelman Report is Negative -- –Choose the gold!

17. Expected Value of Sample Information (EVSI) Recall, P(Positive) =.56 P(Negative) =.44 When positive -- choose Stock with EV = $249 When negative -- choose Gold with EV = $120 Expected Return With Sample Information (ERSI) =.56(249) +.44(120) = $192.50 Expected Return With No Additional Information = EV(Bond) = $130 Expected Value Of Sample Information (EVSI) = ERSI - EV(Bond) = $192.50 - $130 = $62.50

18. Efficiency Efficiency is a measure of the value of the sample information as compared to the theoretical perfect information. It is a number between 0 and 1 given by: Efficiency = EVSI/EVPI For the Jones Investment Model: Efficiency = 62.50/141 =.44

19. Using the Decision Template Bayesian WorksheetResults on Posterior Worksheet Enter Conditional Probabilities

20. Output -- Posterior Analysis Indicator Probabilities Revised Probabilities Optimal Strategy EVSI, EVPI, Efficiency

21. Module C2 Review Expected Value Approach to Decision Making Under Risk EVPI Sample Information –Bayesian Revision of Probabilities –P(Indicator Information) –Strategy –EVSI –Efficiency Use of Decision Template