# Chapter 17 Uncertainty

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Main topics
1.    degree of risk
Chapter 17                                                         2.    decision making under uncertainty
3.    avoiding risk
4.    investing under uncertainty
Uncertainty
ECON252

Degree of risk                                                                  Probability distribution
• probability: number, θ, between 0 and 1 that                                            • relates probability of occurrence to each
indicates likelihood a particular outcome occur                                           possible outcome
• frequency: estimate of probability, θ = n/N, where                                      • first of two following examples is less
n is number of times a particular outcome
occurred during N number of times event occurred
certain
• if we don’t have frequency, may use subjective
probability – informed guess

Figure 17.1 Probability Distribution

(a) Less Certain
Probability, %
(b) More Certain
Probability, %
Expected value example
40                                       40
Probability
distribution     • 2 possible outcomes: rains, does not rain
30                                      30                                           • probabilities are ½ for each outcome
• promoter’s profit is
20                                        20                                              • \$15 with no rain
• -\$5 with rain

10                                        10                                         • promoter’s expected value (“average”)
30%    40%    30%                   EV = [Pr(no rain)×Value(no rain)]+[Pr(rain)×Value (rain)]
10%   20% 40%       20%    10%
= [ ½ × \$15] + [ ½ × (-\$5)] = \$5
0       1      2        3      4
0    1      2        3      4                       Days of rain per month
Days of rain per month

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Variance and standard deviation                    Expected value: another example
• variance: measure of risk                           • 2 possible outcomes: rains, does not rain for a now
• variance = [Pr(no rain) × (Value(no rain - EV)2]      indoor concert
+ [Pr(rain) × (Value (rain) – EV)2]    • probabilities are again ½ for each outcome
= [½ × (\$15 - \$5)2] + [½ × (-\$5 - \$5)2]   • promoter’s profit is now
• \$10 with no rain
= [½ × (\$10)2] + [½ × (-\$10)2] = \$100
• \$0 with rain
• standard deviation = square root of variance
• promoter’s expected value (“average”)
• \$10 in this case                                        EV = [Pr(no rain)×Value(no rain)]+[Pr(rain)×Value (rain)]
= [ ½ × \$10] + [ ½ × (\$0)] = \$5

Variance and standard deviation,
another example
• variance = [Pr(no rain) × (Value(no rain - EV)2]
+ [Pr(rain) × (Value (rain) – EV)2]
= [½ × (\$10 - \$5)2] + [½ × (-\$10 - \$5)2]
= [½ × (\$5)2] + [½ × (\$5)2] = \$25
• standard deviation = square root of variance
• only \$5 in this case

Decision making under uncertainty
• What will the concert promoter choose?
• a rational person might maximize expected
utility: probability-weighted average of
utility from each possible outcome
• promoter’s expected utility from an indoor
concert is
EU = [Pr(no rain) × U(Value(no rain))] + [Pr(rain) × U(Value(rain))]
= [½ × U(\$15)] + [½ × U(-\$5)]

• promoter’s utility increases with wealth
• but at what rate? Increasing decreasing or
constant???

2
Fair bet                                                         Attitudes toward risk
• wager with an expected value of zero                                • someone who is risk averse is unwilling to
• flip a coin for a dollar:                                             make a fair bet
[½ × (1)] + [½ × (-1)] = 0                                 • someone who is risk neutral is indifferent
• someone who is risk preferring wants to
make a fair bet

Figure 17.2 Risk Aversion

Risk aversion                                                             Utility, U

U (Wealth)
c
U (\$70) = 140
0.1U (\$10) + 0.9U (\$70) = 133                                   f
• most people are risk averse: they dislike risk                                       U (\$40) = 120
d

• their utility function is concave to wealth axis: utility          0.5U (\$10) + 0.5U (\$70) =
U (\$26) = 105
e
b
rises with wealth but at a diminishing rate
• they choose the less risky choice if both choices have
the same expected value                                                             U(\$10) = 70
a

• they choose a riskier option only if its expected value is
sufficiently higher than a riskless one
• risk premium: amount that a risk-averse person
would pay to avoid taking a risk
0      10      26        40        64   70 Wealth, \$

Risk averse decision                                           Expected value of Ming Vase
• Irma’s initial wealth is \$40                                        • worth \$10 or \$70 with equal probabilities
• her choice                                                          • expected value (point d):
• she can do nothing: U(\$40) = 120                                                                    \$40 = [½ × \$10] + [½ × \$70]
• she may buy a risky Ming vase                                    • expected utility (point b):
105 = [½ × U(\$10)] + [½ × U(\$70)]

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Figure 17.3a Risk Neutrality

Irma’s risk premium                                            (a) Risk-Neutral Individual

Utility, U

• amount Irma would pay to avoid this risk                                   U (\$70) = 140
c
U (Wealth)

• certain utility from wealth of \$26 is U(\$26) = 105                      0.5U (\$10) +
0.5 U(\$70) =
b
• Irma is indifferent between                                                 U (\$40) = 105

• having the vase
a
• having \$26 with certainty                                                U (\$10) = 70

• thus, Irma’s risk premium is \$14 = \$40 - \$26 to
avoid bearing risk from buying the vase

0   10       40           70         Wealth, \$

Figure 17.3b Risk Preference

Risk-neutral person’s decision                                          (b) Risk-Preferring Individual

Utility, U

• risk-neutral person chooses option with                                    U (\$70) = 140
c
U (Wealth)

highest expected value, because maximizing
expected value maximizes utility                                        0.5 U (\$10) +
0.5U (\$70) = 105
b        e

• utility is linear in wealth                                                  U (\$40) = 82                  d
105 = [½ × U(\$10)] + [½ × U(\$70)]
a
U (\$10) = 70

= [½ × 70] + [½ × 140]
• expected utility = utility with certain wealth
of \$40 (point b)
0   10          40    58 70           Wealth, \$

Risk-preferring person’s decision                                                       Risky jobs
• utility rises with wealth                                   • some occupations have more hazards than
• expected utility from buying vase, 105 at b, is               do others
higher than her certain utility if she does not vase,
82 at d                                                     • in 1995, deaths per 100,000 workers was
• a risk-preferring person is willing to pay for the              • 5 across all industries
right to make a fair bet (negative risk premium)                • 20 for agriculture (35 in crop production)
• Irma’s expected utility from buying vase is same                • 25 for mining
as utility from a certain wealth of \$58, so she’d
pay \$18 for right to “gamble”

4
Risk of workplace homicides per
100,000 workers
Occupation                    Rate                          • Kip Viscusi found workers received a risk
Taxicab driver                22.7                            premium (extra annual earnings) for job hazards
Sheriff-bailiff               10.7                            of \$400 on average in 1969
Police, detective             6.1                           • amount was relatively low because annual risks
incurred by workers were relatively small
Gas station worker            5.9
• in a moderately risky job,
Bartender                     2.3
• danger of dying was about 1 in 10,000
Butcher-meatcutter            1.5                              • risk of a nonfatal injury was about 1 in 100
Fire fighter                  1.3

Value of life                                                Gambling
• given these probabilities, estimated average                 Why would a risk-averse person gamble
job-hazard premium implies that workers                      where the bet is unfair?
placed a value on their lives of about \$1                  • enjoys the game
million                                                    • makes a mistake: can’t calculate odds
• and an implicit value on nonfatal injuries of                correctly
\$10,000                                                    • has Friedman-Savage utility

Application Gambling

Utility, U                          U (Wealth)                             Avoiding risk
e

d                           • just say no: don’t participate in optional
b*
c
d*                           risky activities
a
b                                      • obtain information
• diversify
• risk pooling
• diversification can eliminate risk if two events
are perfectly negatively correlated
W1   W2   W3     W4        W5       Wealth

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Perfectly negatively correlated
• 2 firms compete for government contract
• each has an equal chance of winning
• events are perfectly negatively correlated:
one firm must win and the other must lose
• winner will be worth \$40
• loser will be worth \$10

If buy 1 share of each for \$40                         If buy 2 shares of 1 firm for \$40
• value of stock shares after contract is                  • after contract is awarded, they’re worth \$80 or
awarded is \$50 with certainty                              \$20
• expected value:
• totally diversified: no variance; no risk
\$50 = (½ ¥ \$80) + (½ ¥ \$20)
• variance:
\$900 = [½ ¥ (\$80 - \$50)2] + [½ ¥ (\$20 - \$50)2]
• no diversification (same result if I buy two
stocks that are perfectly positively correlated)

If stocks values are uncorrelated                                       Mutual funds
• each firm has 50% chance of a government               • provide some diversification
contract
• whether a firm gets a contract doesn’t affect
• Standard & Poor’s Composite Index of 500
whether other wins one                                   Stocks (S&P 500)
• expected value                                      • Wilshire 5000 Index Portfolio (actually
\$50 = (¼ ¥ \$80) + (½ ¥ \$50) + (¼ ¥ \$20)
7,200 stocks)
• variance
\$450 = [¼ ¥ (\$80 - \$50)2] + [½ ¥ (\$50 - \$50)2]
+ [¼ ¥ (\$20 - \$50)2]
• buying both results in some diversification

6
Insurance                                           House insurance
• risk-averse people will pay money – risk           •   Scott is risk averse
premium – to avoid risk                            •   wants to insure his \$80 (thousand) house
• world-wide insurance premiums in 1998:             •   25% chance of fire next year
\$2.2 trillion                                      •   if fire occurs, house worth \$40

With no insurance                                            With insurance
• expected value of house is                       • suppose insurance company offers fair insurance
\$70 = (¼ ¥ \$40) + (¾ ¥ \$80)               • lets Scott trade \$1 if no fire for \$3 if fire

• variance                                             • insurance is fair bet because expected value is

\$300 = [¼ ¥ (\$40 - \$70)2] + [¾ ¥ (\$80 - \$70)2]                \$0 = (¼ ¥ [-\$3]) + (¾ ¥ \$1)
• Scott fully insurances: eliminates all risk
• pays \$10 if no fire
• net wealth in both states of nature is \$70

No insurance for terrorism and
Commercial insurance
natural disasters
• is not fair                                        • major natural disasters and terrorism are nondiversifiable
risks because such catastrophic events cause many insured
• available only for diversifiable risks               people to suffer losses at the same time
• more homes have built where damage from storms or
earthquakes is likely, larger potential losses to insurers
from nondiversifiable risks
• insurance companies major losses in 1990s:
• \$12.5 billion for losses in the 1994 Los Angeles earthquake
• \$15.5 billion for Hurricane Andrew in 1992 (total damages were
\$26.5 billion)
• \$3.2 billion for damage from Hurricane Fran in 1995

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Dropping insurance coverage                                               Government steps in
• Farmers Insurance Group reported that it paid out three         • in some areas, state-run pools provide insurance
times as much for the Los Angeles earthquake as it                coverage
collected in earthquake premiums over 30 years.
• Florida Joint Underwriting Association
• insurance companies now refuse to offer hurricane or
• California Earthquake Authority
earthquake insurance in many parts of the country for these
relatively nondiversifiable risks                               • worse deal:
• Nationwide Insurance Company announced in 1996 that it             • these policies extend less protection
was sharply curtailing sales of new policies along the Gulf        • rates are often 3x more than the previously available
of Mexico and the eastern seaboard from Texas to Maine                commercial rates
• a Nationwide official explained, “Prudence requires us to          • require large deductibles
diligently manage our exposure to catastrophic losses.”

Terrorism protection
9/11                                   • insurance companies now view the probability of terrorism
• the terrorist attack of September 11, 2001 inflicted the          as higher than before and are worried that it involves
largest property loss from a single event in history              correlated risk, where several catastrophic events may
• as of a year later, Insurance Information Institute estimated     occur simultaneously
U.S. insured losses at \$40.2 billion—more than the              • immediately after 9/11, insurers added terrorism exclusion
damage from the Los Angeles earthquake and Hurricane              clauses to commercial policies, particularly in aviation and
Andrew combined                                                   real estate
• many other countries suffered sizeable losses from this         • only a few companies still provide such coverage and at
event (including an estimated \$1 billion in insured losses in     very high rates
Japan)                                                          • many firms and local governments stopped buying
• property and casualty insurance industry suffered its first-      terrorism insurance on potential targets, such as San
ever loss for a year in 2001                                      Francisco’s Golden Gate Bridge.

Government response                                          New terrorism insurance law
• many governments around the world provided their                • requires insurers to provide terrorism coverage
airlines with short-term insurance
• commits U.S. government to reimburse insurance
• a group of European insurance and reinsurance companies
companies up to 90% of losses (up to \$100 billion
announced their intention to set up a pool to cover some
types of terrorism risk                                           over 3 years) from foreign terrorist attack
• at the end of 2002, President Bush signed a new terrorism       • President Bush asserted that the lack of terrorism
insurance law                                                     insurance had held up or led to the cancellation of
more than \$15 billion in real estate transactions

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Figure 17.04 Investment Decision Tree with Risk Aversion

Investing under uncertainty                                    (a) Risk-Neutral Owner
High demand
\$200
80%
Invest
EV = \$140
Low demand
• monopoly’s owner has an uncertain payoff                                                                                 20%
–\$100
EV = \$140
this year
Do not invest
• if risk neutral, owner maximizes expected value                                                      \$0

of return
(b) Risk-Averse Owner
• otherwise, owner maximizes his or her                                                                           High demand
U (\$200) = 40
expected utility                                                                     Invest
EU = 32
80%

Low demand
• summarize analysis in decision tree                                                                                      20%
U (– \$100) = 0
EU = 35

Do not invest
U (\$0) = 35

Figure 17.5 Investment Decision Tree with Uncertainty and
Discounting
Investing under uncertainty and
discounting
This year                              Next year
• problem is more complicated if future                                                                                     High demand
R = \$125
returns are uncertain                                                                Invest   EV = \$110
80%

C = \$25 EPV = \$100
• need to calculate expected utility (or value)                                                                              Low demand
R = \$50
20%
and then discount                                          ENVP = \$75

Do not invest
\$0

Figure 17.6 Investment Decision Tree with Advertising

High demand
• future demand is uncertain                                                                                  Advertise     EV =
80%
\$100

– \$50       \$60
• advertising affects demand                                                                                                                Low demand
20%
– \$100
Invest
• suppose risk neutral owner                                                           EV = \$10
High demand
Do not                                     \$100
40%
EV = \$10                                               advertise    EV =
–\$20
Low demand
Do not invest                                                                      – \$100
\$0                                                       60%

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Next
• Externalities and Public Goods

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