Decision Models
description
Transcript of Decision Models
![Page 1: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/1.jpg)
Decision ModelsDecision Models
Making Decisions
Under Risk
![Page 2: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/2.jpg)
Decision Making Under RiskDecision Making Under Risk
• 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 occurring.
![Page 3: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/3.jpg)
The Expected Value ApproachThe Expected Value Approach• Given a set of probabilities for the states of nature,
p1, p2 … etc., for each decision an expected payoff can be calculated by:
pi(payoffi)
• 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.
![Page 4: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/4.jpg)
Expected Value Decision Expected Value Decision • Suppose the broker has offered his own
projections for the probabilities of the states of nature:P(S1) = .2, P(S2) = .3, P(S3) = .3, P(S4) = .1, P(S5) = .1
-$100
-$150-$100$150$200$250
$60
-$200 -$600
$100 $200 $300 $0
S1Lg Rise
S2Sm Rise
S3No Chg.
S4Sm Fall
S5Lg Fall
D1: Gold
D2: Bond
D3: Stock
D4: C/D $60
$500
$60 $60$60
$250 $100
.2 .3 .3 .1 .1Probability
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
![Page 5: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/5.jpg)
Perfect InformationPerfect 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 perfect informationinformation
• Then when you knew S1 was going to occur, you would make the best decision for S1 (Stock = $500). This would happen p1 = .2 of the time.
• When you knew S2 was going to occur, you would make the best decision for S2 (Stock = $250). This would happen p2 = .3 of the time.
• And so forth
![Page 6: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/6.jpg)
Expected Value of Perfect Expected Value of Perfect Information (EVPI)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.
![Page 7: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/7.jpg)
Calculating the EVPICalculating the EVPI
-$100
-$150-$100$150$200$250
$60
-$200 -$600
$100 $200 $300 $0
S1Lg Rise
S2Sm Rise
S3No Chg.
S4Sm Fall
S5Lg Fall
D1: Gold
D2: Bond
D3: Stock
D4: C/D $60
$500
$60 $60$60
$250 $100
.2 .3 .3 .1 .1Probability
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
![Page 8: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/8.jpg)
Using the Decision TemplateUsing the Decision Template
EnterProbabilities
Expected Value DecisionEVPI
![Page 9: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/9.jpg)
Sample InformationSample 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(S1) = .2, P(S2) = .3, P(S3) = .3, P(S4) = .1, P(S5) = .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).
![Page 10: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/10.jpg)
Sample Information ApproachSample 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.
![Page 11: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/11.jpg)
Example -- Samuelman ForecastExample -- 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) = .7P(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
![Page 12: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/12.jpg)
Bayesian ProbabilitiesBayesian ProbabilitiesGiven a Positive ForecastGiven 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) (.30)
(.30)(.40) (.10)(0) (.10) = .56= .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 = .286.286(.70) (.30) /.56 = .375.375(.50) (.30) /.56 = .268.268(.40) (.10) /.56 = .071.071
(0) (.10) /.56 = 00
(.50)
![Page 13: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/13.jpg)
Best Decision With Positive Best Decision With Positive ForecastForecast
-$100
-$150-$100$150$200$250
$60
-$200 -$600
$100 $200 $300 $0
S1Lg Rise
S2Sm Rise
S3No Chg.
S4Sm Fall
S5Lg Fall
D1: Gold
D2: Bond
D3: Stock
D4: C/D $60
$500
$60 $60$60
$250 $100
.286 .375 .268 .071 0Revised
Probability
Expected Value
$84
$180
$249
$60
Highest With Positive Forecast -- Choose D3 - Stock
When Samuelman predicts “positive” -- Choose the Stock!
![Page 14: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/14.jpg)
Bayesian ProbabilitiesBayesian ProbabilitiesGiven a Negative ForecastGiven 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) (.20)(.30) (.30)
(.30)(.60) (.10)(1) (.10) = .44= .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 = .091.091(.30) (.30) /.44 = .205.205(.50) (.30) /.44 = .341.341(.60) (.10) /.44 = .136.136
(1) (.10) /.44 = .227.227
(.50)
![Page 15: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/15.jpg)
Best Decision With Negative Best Decision With Negative ForecastForecast
-$100
-$150-$100$150$200$250
$60
-$200 -$600
$100 $200 $300 $0
S1Lg Rise
S2Sm Rise
S3No Chg.
S4Sm Fall
S5Lg Fall
D1: Gold
D2: Bond
D3: Stock
D4: C/D $60
$500
$60 $60$60
$250 $100
.091 .205 .341 .136 .227Revised
Probability
Expected Value
$120
$ 67
-$33
$60
Highest With Negative Forecast -- Choose D1 - Gold
When Samuelman predicts “negative” -- Choose Gold!
![Page 16: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/16.jpg)
Strategy With Sample Strategy With Sample InformationInformation
• If the Samuelman Report is PositivePositive --
–Choose the stock!Choose the stock!
• If the Samuelman Report is NegativeNegative --
–Choose the gold!Choose the gold!
![Page 17: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/17.jpg)
Expected Value of Sample Expected Value of Sample Information (EVSI)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) =(ERSI) =.56 (249) + .44 (120) = $192.50= $192.50
Expected Return With No Additional Information =EV(Bond) = $130= $130
Expected Value Of Sample Information (EVSI) =(EVSI) =ERSI - EV(Bond) = $192.50 - $130 = $62.50= $62.50
![Page 18: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/18.jpg)
EfficiencyEfficiency
• 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/EVPIEfficiency = EVSI/EVPI
• For the Jones Investment Model:
Efficiency = 62.50/141 = .44Efficiency = 62.50/141 = .44
![Page 19: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/19.jpg)
Using the Decision TemplateUsing the Decision Template
Bayesian Worksheet
Results on Posterior Worksheet
Enter Conditional Probabilities
![Page 20: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/20.jpg)
Output -- Posterior AnalysisOutput -- Posterior Analysis
Indicator ProbabilitiesRevised ProbabilitiesOptimal StrategyEVSI, EVPI, Efficiency
![Page 21: Decision Models](https://reader035.fdocuments.in/reader035/viewer/2022062500/5681592c550346895dc65791/html5/thumbnails/21.jpg)
ReviewReview
• 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