Chapter 17
Uncertainty
Topics
• 
• 
• 
• 
• 
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Degree of Risk.
Decision Making Under Uncertainty.
Avoiding Risk.
Investing Under Uncertainty.
Behavioral Economics of Risk.
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Risk
•  Risk - situation in which the likelihood of
each possible outcome is known or can
be estimated and no single possible
outcome is certain to occur.
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Probability
•  A probability is a number between 0 and 1
that indicates the likelihood that a
particular outcome will occur.
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Frequency
•  Let n be the number of times one
particular outcome occurred during the N
total number of times an event occurred.
•  We set our estimate of the probability, θ,
equal to the frequency:
θ = n/N.
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Frequency (cont.)
•  A house either burns or does not burn.
•  If n = 13 similar houses burned in your
neighborhood of N = 1,000 homes last
year, you might estimate the probability
that your house will burn this year as
θ = 13/1,000 = 1.3%.
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Subjective Probability
•  Subjective probability - our best
estimate of the likelihood that an outcome
will occur
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Probability Distribution
•  A probability distribution relates the
probability of occurrence to each possible
outcome.
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Figure 17.1 Probability Distribution
30
Probability
distribution
40
30
20
10
10
0
20%
1
40%
20%
10%
Probability
distribution
20
10%
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40
(b) More Cer tain
Probability, %
Probability, %
(a) Less Cer tain
30%
2
3
4
Days of rain per month
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0
1
40%
30%
2
3
4
Days of rain per month
Probability Distribution (cont.)
•  Mutually exclusive – when only one of
the outcomes can occur at a given time.
•  Exhaustive – when no other outcomes
than those listed are possible.
w Where outcomes are mutually exclusive and
exhaustive, exactly one of these outcomes
will occur with certainty, and the probabilities
must add up to 100%.
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Expected Value
•  Gregg, a promoter, schedules an outdoor
concert for tomorrow.
w How much money he’ll make depends on the
weather.
•  If it doesn’t rain, his profit or value from the
concert is V = $15.
•  If it rains, he’ll have to cancel the concert and he’ll
lose V = −$5, which he must pay the band.
•  He knows that the weather department forecasts a
50% chance of rain.
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Expected Value (cont.)
•  The expected value, EV, is the value of
each possible outcome times the
probability of that outcome:
EV = [ Pr (no rain) × Value(no rain) ] + [ Pr (rain) × Value(rain) ]
⎡1
⎤ ⎡1
⎤
= ⎢ × $15⎥ + ⎢ × ( −$5 )⎥ = $5
⎣2
⎦ ⎣2
⎦
w  where Pr is the probability of an outcome,
•  so Pr(rain) is the “probability that rain occurs.”
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Solved Problem 17.1
•  How much more would Gregg expect to
earn if he knew that he would obtain
perfect information about the probability of
rain far enough before the concert that he
could book the band only if needed? How
much does he gain from having this
perfect information?
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Variance and Standard Deviation
•  Variance (σ2) - measures the spread of
the probability distribution.
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Variance and Standard Deviation
•  Standard deviation (σ)- the square root
of the variance.
•  Holding the expected value constant, the
smaller the standard deviation (or
variance), the smaller the risk.
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Table 17.1 Variance and Standard
Deviation: Measures of Risk
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Decision Making Under Uncertainty
•  Suppose that Gregg strikes a new
agreement with the band by which he
pays only if the weather is good and the
concert is held. What would be the
expected value and variance for this
case?
w  EV = $7.50
w  s2 = $56.25
w  s = $7.50
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Decision Making Under Uncertainty
(cont.)
•  Suppose that his choice is between the indoor
concert and an outdoor concert from which he
earns $100,015.50 if it doesn’t rain and loses
$100,005 if it rains. Again, calculate the new
expected value and standard deviation.
•  EV = $5.25
•  s = $100,010.25
w  So he loses $100,005 with a 50% probability.
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Expected Utility
•  Expected utility (EU) - the probability-weighted
average of the utility from each possible
outcome.
w  For example, Gregg’s EU from the outdoor concert
is:
EU = ⎡⎣ Pr (no rain) × U ( Value(no rain) )⎤⎦ + ⎡⎣ Pr (rain) × U ( Value(rain) )⎤⎦
⎡1
⎤ ⎡1
⎤
= ⎢ × U ( $15 )⎥ + ⎢ × U ( −$5 )⎥ ,
⎣2
⎦ ⎣2
⎦
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Expected Utility (cont.)
•  Fair bet - a wager with an expected value of
zero.
•  Example: you pay a dollar if a flipped
coin comes up heads and receive a
dollar if it comes up tails. Because you
expect to win half the time and lose half
the time, the EV is:
⎡1
⎤ ⎡1
⎤
= ⎢ × ( −$1)⎥ + ⎢ × ( $1)⎥ = 0.
⎣2
⎦ ⎣2
⎦
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Expected Utility (cont.)
•  Risk averse - unwilling to make a fair bet.
•  Risk neutral - indifferent about making a
fair bet.
•  Risk preferring - willing to make a fair
bet.
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Expected Utility (cont.)
•  Irma, who is risk averse, makes a choice under
uncertainty. She has an initial wealth of $40 and
has two options:
w  nothing and keep the $40.
w  buy a vase.
•  Her wealth is:
w  $70 if the vase is a Ming and
w  $10 if it is an imitation.
•  Irma’s subjective probability is 50% that it is a
genuine Ming vase.
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Utility, U
Figure 17.2 Risk Aversion
U(
Wealth)
c
U($40)
= 120
0.5U
($10)
+ 0.5U
($70)
=
U($26)
= 105
U($70) = 140
0.1U($10) + 0.9U($70) = 133
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a
b
10
Risk premium
26
40
0
e
ì
ï
í
ï
î
U($10) = 70
d
f
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64
70
Wealth,
$
Risk Aversion
A person whose utility function is concave
picks the less risky choice if both choices
have the same expected value.
•  Risk premium - the amount that a riskaverse person would pay to avoid taking a
risk
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Solved Problem 17.2
•  Suppose that Irma’s subjective probability
is 90% that the vase is a Ming. What is
her expected wealth if she buys the vase?
What is her expected utility? Does she
buy the vase?
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Risk Neutrality
•  Someone who is risk neutral has a
constant marginal utility of wealth:
w Each extra dollar of wealth raises utility by the
same amount as the previous dollar.
•  the utility curve is a straight line in a utility and
wealth graph.
A risk-neutral person chooses the option
with the highest expected value, because
maximizing expected value maximizes
utility.
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Risk Preference
•  An individual with an increasing marginal
utility of wealth is risk preferring:
w Willing to take a fair bet.
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Figure 17.3 Risk Neutrality and
Risk Preference
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Application Gambling
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Avoiding Risk
• 
• 
• 
• 
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Just Say No
Obtain Information
Diversify
Insure
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Correlation and Diversification
•  The extent to which diversification
reduces risk depends on the degree to
which various events are correlated over
states of nature.
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Correlation and Diversification (cont.)
•  If you know that the first event occurs,
w you know that the probability that the second
event occurs is lower if the events are
negatively correlated
w and higher if the events are positively
correlated.
•  The outcomes are independent or
uncorrelated if knowing whether the first
event occurs tells you nothing about the
probability that the second event occurs.
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Correlation and Diversification (cont.)
•  Diversification can eliminate risk if two
events are perfectly negatively correlated.
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Correlation and Diversification (cont.)
•  Suppose that two firms are competing for
a government contract and have an equal
chance of winning.
•  Because only one firm can win, the other
must lose, so the two events are perfectly
negatively correlated.
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Correlation and Diversification (cont.)
•  You can buy a share of stock in either firm
for $20.
•  The stock of the firm that wins the
contract will be worth $40,
•  the stock of the loser will be worth $10.
•  If you buy two shares of the same
company, your shares are going to be
worth either $80 or $20 after the contract
is awarded.
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Correlation and Diversification (cont.)
•  Their expected value is:
$50 = (1/2 x $80) +(1/2 x $20)
•  And the variance:
2⎤
2⎤
⎡1
⎡1
$900 = ⎢ × ( $80 − $50 ) ⎥ + ⎢ × ( $20 − $50 ) ⎥ .
⎣2
⎦ ⎣2
⎦
•  However, if you buy one share of each,
your two shares will be worth $50 no
matter which firm wins, and the variance
is zero.
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Correlation and Diversification (cont.)
The more negatively correlated two events
are, the more diversification reduces risk.
Diversification does not reduce risk if two
events are perfectly positively correlated.
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APPLICATION Mutual Funds
•  Mutual fund - issued by a company that
buys stocks in many other companies.
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Insure
•  Because Scott is risk averse, he wants to
insure his house, which is worth $80
(thousand).
•  There is a 25% probability that his house will
burn next year.
•  If a fire occurs, the house will be worth only
$40.
•  With no insurance, the expected value of his
house is: 1
3
⎛
⎞ ⎛
⎞
⎜ × $40 ⎟ + ⎜ × $80 ⎟ = $70.
⎝4
⎠ ⎝4
⎠
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Insure (cont.)
•  The variance of the value of his house is
2⎤
2⎤
⎡1
⎡3
⎢⎣ 4 × ( $40 − $70 ) ⎥⎦ + ⎢⎣ 4 × ( $80 − $70 ) ⎥⎦ = $300.
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Insure (cont.)
•  Fair insurance - a bet between an insurer and a
policyholder in which the value of the bet to the
policyholder is zero.
•  The insurance company offers to let Scott trade $1 in
the good state of nature (no fire) for $3 in the bad state
of nature (fire).
w  This insurance is fair because the expected value of this
insurance to Scott is zero:
⎡1
⎤ ⎡3
⎤
×
$3
+
×
−
$1
=
$0.
(
)
⎢⎣ 4
⎥⎦ ⎢⎣ 4
⎥⎦
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Insure (cont.)
•  Because Scott is risk averse, he fully
insures by buying enough insurance to
eliminate his risk altogether.
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Insure (cont.)
•  Scott
w  pays the insurance company $10 in the good state of nature
and
w  Receives $30 in the bad state.
•  In the good state,
w  he has a house worth $80 less the $10 he pays the insurance
company, for a net wealth of $70.
w  If the fire occurs, he has a house worth $40 plus a payment
from the insurance company of $30, for a net wealth, again, of
$70.
•  Scott’s expected value with fair insurance, $70, is the
same as his expected value without insurance.
•  The variance he faces drops from $300 without
insurance to $0 with insurance.
•  Scott is better off with insurance because he has the
same expected value and faces no risk.
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Solved Problem 17.3
•  The local government assesses a
property tax of $4 (thousand) on Scott’s
house. If the tax is collected whether or
not the house burns, how much fair
insurance does Scott buy? If the tax is
collected only if the house does not burn,
how much fair insurance does Scott buy?
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Risk-Neutral Investing
•  Chris, the owner of the monopoly, is risk
neutral.
•  She maximizes her expected utility by
making the investment only if the
expected value of the return from the
investment is positive.
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Figure 17.4a Investment Decision
Tree with Risk Aversion
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Risk-Averse Investing
•  Ken, who is risk averse, faces the same
decision as Chris.
•  Ken invests in the new store if his
expected utility from investing is greater
than his certain utility from not investing.
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Figure 17.4b Investment Decision
Tree with Risk Aversion (cont.)
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Investing with Uncertainty and
Discounting
•  How does this rule change if the returns
are uncertain?
•  A risk-neutral person chooses to invest if
the expected net present value is positive.
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Figure 17.5 Investment Decision Tree
with Uncertainty and Discounting
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Solved Problem 17.4
•  Gautam, who is risk neutral, is considering
whether to invest in a new store, as the figure
shows. After investing, he can increase the
probability that demand will be high at the new
store by advertising at a cost of $50. Should he
invest?
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Solved Problem 17.4
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Gambler’s Fallacy
•  Gambler’s fallacy - arises from the false
belief that past events affect current,
independent outcomes.
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Overconfidence
•  A common explanation for why some
people engage in gambles that the rest of
us avoid like the plague is that these
gamblers are overconfident.
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Certainty Effect
•  First, a group of subjects were asked to choose
between two options:
w  Option A. You receive $4,000 with probability 80%
and $0 with probability 20%.
w  Option B. You receive $3,000 with certainty.
w  The vast majority, 80%, chose the certain outcome,
B.
•  Then, the subjects were given another set of
options:
w  Option C. You receive $4,000 with probability 20%
and $0 with probability 80%.
w  Option D. You receive $3,000 with probability 25%
and $0 with probability 75%.
•  Now, 65% prefer C.
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Certainty Effect (cont.)
•  Kahneman and Tversky found that over
half the respondents violated expected
utility theory by choosing B in the first
experiment and C in the second one.
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Framing
•  The United States expects an unusual disease
(e.g., avian flu) to kill 600 people.
•  The government is considering two alternative
programs to combat the disease.
•  The “exact scientific estimates” of the
consequences of these programs are:
w  If Program A is adopted, 200 people will be saved.
w  If Program B is adopted, there is a 1 3 probability that
600 people will be saved and a 2/3 probability that no
one will be saved.
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Framing (cont.)
•  When college students were asked to
choose, 72% opted for the certain gains of
Program A over the possibly larger but
riskier gains of Program B.
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Framing (cont.)
•  A second group of students was asked to
choose between an alternative pair of programs,
and were told:
w  If Program C is adopted, 400 people will die.
w  If Program D is adopted, there is a 1/3 probability
that no one will die, and a 2/3 probability that 600
people will die.
•  When faced with this choice, 78% chose the
larger but uncertain losses of Program D over
the certain losses of Program C.
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Prospect Theory
•  Prospect theory - is an alternative theory
of decision-making under uncertainty that
can explain some of the choices people
make that are inconsistent with expected
utility theory.
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Figure 17.6 Prospect Theory Value
Function
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