decisions -- under risk

7/27/2019 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


8/21

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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