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Decision 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.

The 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.

Expected Value Decision

Probability

.2

.3

S1

Lg Rise

.3

S2

Sm Rise

.1

S3

No Chg.

.1

S4

Sm Fall

S5

Lg Fall

D1: Gold

-$100

$100

$200

$300

$0

D2: Bond

$250

$200

$150

-$100

-$150

D3: Stock

$500

$250

$100

-$200

-$600

D4: C/D

$60

$60

$60

$60

$60

Highest -- Choose D2 - Bond

- 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

Expected Value

.2(-100)+.3(100)+.3(200)+.1(300)+.1(0)

$100

.2(250)+.3(200)+.3(150)

+.1(-100)+.1(-150)

$130

.2(500)+.3(250)+.3(100)

+.1(-200)+.1(-600)

$125

$60

.2(60)+.3(60)+.3(60)

+.1(60)+.1(60)

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 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

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.

Calculating the EVPI

Probability

.2

.3

S1

Lg Rise

.3

S2

Sm Rise

.1

S3

No Chg.

.1

S4

Sm Fall

S5

Lg Fall

D1: Gold

-$100

$100

$200

$300

$0

D2: Bond

$250

$200

$150

-$100

-$150

D3: Stock

$500

$250

$100

-$200

-$600

D4: C/D

$60

$60

$60

$60

$60

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

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(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).

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.

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

Bayesian ProbabilitiesGiven a Positive Forecast

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)

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)

(.80)

(.20)

(.70)

(.30)

(.50)

(.30)

(.40)

(.10)

= .56

(0)

(.10)

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

Best Decision With Positive Forecast

RevisedProbability

.286

.375

.268

.071

0

S1

Lg Rise

S2

Sm Rise

S3

No Chg.

S4

Sm Fall

S5

Lg Fall

D1: Gold

-$100

$100

$200

$300

$0

D2: Bond

$250

$200

$150

-$100

-$150

D3: Stock

$500

$250

$100

-$200

-$600

D4: C/D

$60

$60

$60

$60

$60

Highest With Positive Forecast -- Choose D3 - Stock

Expected Value

$84

$180

$249

$60

When Samuelman predicts “positive” -- Choose the Stock!

Bayesian ProbabilitiesGiven a Negative Forecast

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)

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)

(.20)

(.20)

(.30)

(.30)

(.50)

(.30)

(.60)

(.10)

= .44

(1)

(.10)

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

Best Decision With Negative Forecast

RevisedProbability

.091

.205

.341

.136

.227

S1

Lg Rise

S2

Sm Rise

S3

No Chg.

S4

Sm Fall

S5

Lg Fall

D1: Gold

-$100

$100

$200

$300

$0

D2: Bond

$250

$200

$150

-$100

-$150

D3: Stock

$500

$250

$100

-$200

-$600

Highest With Negative Forecast -- Choose D1 - Gold

D4: C/D

$60

$60

$60

$60

$60

Expected Value

$120

$ 67

-$33

$60

When Samuelman predicts “negative” -- Choose Gold!

Strategy With Sample Information

- If the Samuelman Report is Positive --
- Choose the stock!

- If the Samuelman Report is Negative --
- Choose the gold!

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

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

Using the Decision Template

Enter Conditional Probabilities

Bayesian Worksheet

Results on Posterior Worksheet

Output -- Posterior Analysis

Indicator ProbabilitiesRevised Probabilities

Optimal Strategy

EVSI, EVPI, Efficiency

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

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