decisions -- under risk
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Decision Models
Making DecisionsUnder Risk
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Decision Making Under Risk
When doing decision making underuncertainty, 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 thelikelihood of each state of natureoccurring.
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The Expected Value Approach Given a set of probabilities for the states of
nature, p1
, p2
etc., for each decision anexpected payoff can be calculated by:
pi(payoffi) If this is a decision that will be repeated over
and over again, the decision with the highestexpected payoff should be the one selected tomaximize total expected payoff.
But if this is a one-time decision, perhaps therisk of losing much money may be too great --thus the expected payoff is just another pieceof information to be considered by the decisionmaker.
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Expected Value Decision
Suppose the broker has offered his ownprojections for the probabilities of the statesof 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
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 $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
H ighest -- Choose D2 - Bond
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Perfect Information
Although the states of nature are assumed to
occur with the previous probabilities, supposeyou knew, each time which state of naturewould occur -- i.e. you had perfect information
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
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Expected Value of PerfectInformation (EVPI)
The expected value of perfectinformation (EVPI) is the gain in
value from knowing for sure whichstate of nature will occur when,versus only knowing theprobabilities.
It is the upper bound on the value ofany additional information.
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Calculating 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: StockD4: 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) = $271Expected Return With No Additional Information =
EV(Bond) = $130
Expected Value Of Perfect Information(EVPI) =
ERPI - EV(Bond) = $271 - $130 = $141
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Using the Decision Template
Enter
Probabilities
Expected Value Decision
EVPI
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Sample Information
One never really has perfect information,
but can gather additional information, getexpert advice, etc. that can indicate whichstate of nature is likely to occur each time.
The states of nature still occur, in the longrun 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 eachpossibility of the indicator information
We want to know the value of this sample
information (EVSI).
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Sample Information Approach
Given the outcome of the sampleinformation, we revise the probabilities ofthe states of nature occurring (using
Bayesian analysis). Then we repeat the expected value
approach (using these revised
probabilities) to see which decision isoptimal given each possible value of thesample information.
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Example -- Samuelman Forecast
Noted economist Milton Samuelman gives
an economic forecast indicating eitherPositive or Negative economic growth in thecoming year.
Using a relative frequency approach basedon 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) = .3P(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
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Bayesian ProbabilitiesGiven 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
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
(.50)
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Best Decision With PositiveForecast
-$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 $60$60
$250 $100
.286 .375 .268 .071 0
Revised
Probability
Expected Value
$84
$180
$249
$60
H ighest With Positive Forecast -- Choose D3 - Stock
When Samuelman predicts positive-- Choose the Stock!
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Bayesian ProbabilitiesGiven 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
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
(.50)
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Best Decision With NegativeForecast
-$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 $60$60
$250 $100
.091 .205 .341 .136 .227
Revised
Probability
Expected Value
$120
$ 67
-$33
$60
H ighest With Negative Forecast -- Choose D1 - Gold
When Samuelman predicts negative-- Choose Gold!
St t With S l
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Strategy With SampleInformation
If the Samuelman Report is Positive --Choose the stock!
If the Samuelman Report is Negative --
Choose the gold!
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Expected Value of SampleInformation (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) = $192.50Expected Return With No Additional Information =
EV(Bond) = $130
Expected Value Of Sample Information(EVSI) =
ERSI - EV(Bond) = $192.50 - $130 = $62.50
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Efficiency
Efficiency is a measure of the value of thesample information as compared to thetheoretical 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
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Using the Decision Template
Bayesian Worksheet
Results on Posterior Worksheet
Enter Conditional Probabilities
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Output -- Posterior Analysis
Indicator ProbabilitiesRevised Probabilities
Optimal Strategy
EVSI, EVPI, Efficiency
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Review
Expected Value Approach toDecision Making Under Risk
EVPI
Sample Information Bayesian Revision of Probabilities
P(Indicator Information)
Strategy EVSI
Efficiency
Use of Decision Template
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