Module C2

1
Module C2

• Decision Models
• Decision Making Under Risk

2
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 occuring

3
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.

4
Expected Value Decision for Tom Brown

• 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
.2(60).3(60).3(60) .1(60).1(60)
60
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 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 p1 .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

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
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(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).

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(Positivelarge rise) .8 P(Negativelarge
  rise) .2
• P(Positivesmall rise) .7 P(Negativesmall
  rise) .3
• P(Positiveno change) .5 P(Negativeno
  change) .5
• P(Positivesmall fall) .4 P(Negativesmall
  fall) .6
• P(Positivelarge fall) 0 P(Negativelarge
  fall) 1

12
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(PositiveLarge Rise)P(Large
Rise) P(PositiveSmall Rise) P(Small Rise)
P(PositiveNo Change)P(No Change)
P(PositiveSmall Fall) P(Small Fall)
P(PositiveLarge Fall) P(Large Fall)
(.80)
(.20)

(.70)
(.30)
(.50)
(.30)
(.40)
(.10)
(0)
.56
(.10)
P(Large RisePos) P(PosLg. Rise)P(Lg.
Rise)/P(Pos) P(Small RisePos) P(PosSm.
Rise)P(Sm. Rise)/P(Pos) P(No ChangePos)
P(PosNo Chg.)P(No Chg.)/P(Pos) P(Small FallPos)
P(PosSm. Fall)P(Sm. Fall)/P(Pos) P(Large
FallPos) P(PosLg. 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
Expected Value
84
180
249
60
When Samuelman predicts positive -- Choose the
Stock!
14
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(NegativeLarge Rise)P(Large
Rise) P(NegativeSmall Rise) P(Small Rise)
P(NegativeNo Change)P(No Change)
P(NegativeSmall Fall) P(Small Fall)
P(NegativeLarge Fall) P(Large Fall)
(.20)
(.20)

(.30)
(.30)
(.50)
(.30)
(.60)
(.10)
(1)
.44
(.10)
P(Large RiseNeg) P(NegLg. Rise)P(Lg.
Rise)/P(Neg) P(Small RiseNeg) P(NegSm.
Rise)P(Sm. Rise)/P(Neg) P(No ChangeNeg)
P(NegNo Chg.)P(No Chg.)/P(Neg) P(Small FallNeg)
P(NegSm. Fall)P(Sm. Fall)/P(Neg) P(Large
FallNeg) P(NegLg. 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
Expected Value
120
67
-33
60
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
20
Output -- Posterior Analysis
Indicator ProbabilitiesRevised
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