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Part 3
Uncertainty and
Strategy
© 2006 Thomson Learning/South-Western
Chapter 5
Uncertainty
© 2006 Thomson Learning/South-Western
Probability

Probability of an event happening;
relative frequency with which an event
occurs


3
Probability of “heads” coming up on flip of fair
coin is ½.
That is, when coin is flipped many times, we
can expect “heads” to come up in
approximately one-half of flips.
Expected Value


4
Expected value of game with a number of uncertain
outcomes: size of prize that player will win on average.
On a single flip of a coin, Jones pays Smith $1 (X1 =
+$1) if a tail comes up and Smith will pay Jones $1 (X2
= -$1) if a head comes up, the expected value of game
for both players is
1
1
1
1
X 1  X 2  ( $1)  ( $1)  0.
2
2
2
2
Expected Value

If game changes so that, from Smith’s point of
view, X1 = $10, and X2 = -$1, expected value for
Smith would be:
1
1
1
1
X 1  X 2  ($10)  ( $1)  $4.50.
2
2
2
2


5
Because Smith would stand to win $4.50 on
average, she might be willing to pay Jones up
to this amount to play.
Fair games are games that cost precisely their
expected value.
Risk Aversion



6
When people face risky but fair situations,
they will usually choose not to participate.
Risk aversion is tendency for people to
refuse to accept fair games.
Swiss mathematician Daniel Bernoulli
theorized that not only the monetary payoff of
a game matters to people; expected utility
from the game’s prizes also affects people’s
willingness to play.
Diminishing Marginal Utility
7

Bernoulli assumed that utility associated
with payoffs in risky situation increases
less rapidly than dollar value of payoffs.

Extra (marginal) utility obtained from
winning an extra dollar in prize money is
assumed to decline as additional dollars
are won.
Diminishing Marginal Utility


Fig. 5.1 reflects diminishing marginal
utility; shows utility associated with
possible prizes (or incomes) from $0 to
$50,000.
Concave shape of curve reflects
assumed diminishing marginal utility.

8
Gain in utility due to increased income from
$1000 to $2000 exceeds gain from
$49,000 to $50,000.
Graphical Analysis of Risk Aversion

Figure 5-1 shows person with three
options. Contender may:


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9
retain current income level ($35,000) without
taking any risk;
take fair bet with 50-50 chance of winning or
losing $5,000;
take fair bet with a 50-50 chance of winning
or losing $15,000.
FIGURE 5-1: Risk Aversion
Utility
U
0
10
20
30
33 35
40
50
Income
(thousands
of dollars)
A graphical Analysis of Risk
Aversion


Current $35,000 provides utility of U3.
Utility of $5,000 bet is average of utility of
$40,000 (if player wins) and utility of
$30,000 (if player loses).


11
Average utility is U2 < U3.
The utility (U1 < U2) of the $5000 bet is
average of utility of winning ($50,000) and
losing ($20,000).
FIGURE 5-1: Risk Aversion
Utility
U
U3
U2
U1
0
12
20
30
33
35
40
50
Income
(thousands
of dollars)
Willingness to Pay to Avoid Risk

With risk aversion and equal expected
values ($35,000 for three options in
Figure 5-1), contender will prefer riskfree incomes to risky incomes that offer
less utility.

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13
Figure 5-1,risk-free income of $33,000
provides same utility as $5,000 gamble.
Player would pay up to $2,000 to avoid
taking risk.
Methods of Reducing Risk:
Insurance


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14
Figure 5-2 shows motive for buying
insurance.
Assume that during next year person
with $10,000 current income faces 50
percent chance of incurring $4,000 in
unexpected medical bills.
Without insurance, person’s utility would
be U1--utility of average of $6000 and
$10,000.
FIGURE 5-2: Insurance Reduces
Risk
Utility
U
U1
0
15
20
25
35
Income
(thousands
of dollars)
Fair Insurance

Fair insurance: premium equals
expected value of loss.

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16
Figure 5-2, fair insurance would cost
$7,500--expected value of what insurance
companies would have to pay each year in
health claims.
Would guarantee income of $27,500--yield
utility of U2
FIGURE 15.2: Insurance Reduces
Risk
Utility
U
U2
U1
0
17
20
25
27.5
35
Income
(thousands
of dollars)
Unfair Insurance



18
Since insurance companies have costs
beyond paying benefits, they can not sell
insurance at actuarially fair premiums.
Figure 5-2: client would be willing to pay up
to $10,000 for health insurance since
$25,000 of risk-free income yields as much
utility (U1) as going without any insurance.
$12,000 premium would reduce utility to
U 0.
FIGURE 5-2: Insurance Reduces
Risk
Utility
U
U2
U1
U0
0
19
20 23
25
27.5
35
Income
(thousands
of dollars)
Uninsurable Risks

Some risks so unique or difficult to
evaluate that insurers unable to set
premium rates--risks become uninsurable.

If events so infrequent or unpredictable
(such as wars, “Acts of God” etc.) insurers
have no basis for establishing premiums.
20
Methods of Reducing Risk:
Diversification


Diversification: economic version of “Don’t
put all your eggs in one basket.”
Diversification spreads risk among
several options rather than choosing only
one.

21
Suitably spreading risk may increase utility
above that obtain by a single transaction.
Diversification


Figure 5-3 shows income utility for
individual with current income of $35,000
who must invest $15,000 in risky assets.
Assume only two such assets: shares of
company A or company B stock

22
Each company’s stock costs $1, but value will
increase to $2 if company does well during
next year.
Diversification



23
If either company does poorly, its stock will be
worthless.
Each company has a 50-50 chance of doing
well.
If A and B are unrelated to one another,
holding both stocks will reduce investor’s
risks.
Diversification

Investing in 15,000 shares of company A
yields a 50 percent chance of having
$50,000 and a 50 percent chance of
having $20,000.


24
Yields a utility level of U1.
If person invests in 7,500 shares of each
company, faces four possible outcomes
shown in Table 5-1.
FIGURE 5-3: Diversification
Reduces Risk
Utility
U
U1
0
25
20
35
50
Income
(thousands
of dollars)
TABLE 5-1: Possible Outcomes from
Investing in Two Companies
Company A’s
Performance
26
Poor
Good
Company B’s Performance
Poor
Good
$20,000
$35,000
35,000
50,000
Diversification

Each of four outcomes is equally likely;
with half of cases, investor ends up with
original $35,000.

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
27
Diversification strategy, while it still has an
expected value of $35,000, has less risk.
Figure 5-3, point C represent when B does poorly
and D represent when B does well.
Point E, (the average of C and D) results from
diversification and yields utility U2 > U1.
FIGURE 5-3: Diversification
Reduces Risk
Utility
D
U2
U1
U
E
C
0
28
20
35
50
Income
(thousands
of dollars)
Attributes of Options
 Specification of underlying transaction
 Definition of period during which option
may be exercised
 Price of option
29
Value of Underlying Transaction Affects Option Value


Because transaction that underlies an
option will occur in the future, underlying
transaction’s value subject to many
uncertainties.
The value of underlying transaction in
option has two general dimensions

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30
Expected value of transaction
Variability of value of transaction.
Duration of Option Affects Its Value

The longer an option lasts, the more
valuable it is.

Intuitively, more time you have to take
advantage of flexibility an option offers,
more likely it is that you will want to do so.
31
Investors’ Market Options

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

32
Figure 5-4 shows simplified illustration of
market options open to a would-be investor in
financial assets.
Points on figure represent options available.
For example, point A represents a risk-free asset
such as money in a checking account.
Asset B represents relatively risky stock.
All other points on Figure 5-4 represent risks
and returns associated with assets that investors
might buy.
FIGURE 5-4: Market Options for
Investors
Market Line
Annual
return
C
B
A
Risk
33
Investors’ Market Options



34
Investors like high annual returns but dislike risk,
so they will choose to hold combinations of
these available assets that lie on their
“northwest” periphery.
By mixing various risky assets with risk-free
asset (A), they can choose any point along the
line AC.
Market line: shows possible combinations of
annual returns and risk that investors can
achieve by taking advantage of what market the
offers.
Choices by Individual Investors



35
The market line in Figure 5-4 provides a
constraint on the options that financial
markets provide to individual investors.
These investors then choose among the
available options on the basis of their own
attitudes toward risk.
This process is illustrated in Figure 5-5.
Choices by Individual Investors


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36
The figure shows a typical indifference
curve for three different types of investors.
The three investors illustrated in Figure 5-5
have different attitudes toward risk.
Investor I has a very low tolerance for risk.
He will opt for mix of investments that
include a lot of the risk-free option (point L).
Choices by Individual Investors


37
Investor II has a moderate toleration for
risk. She will opt for a combination of
assets that are reasonably representative
of the overall market (M).
Finally, investor III is a real speculator.
She will accept a risky combination of
assets (N) – more risky than the overall
market.
FIGURE 5-5: Choices by Individual
Investors
UIII
UII
Annual
return
Market Line
N
UI
M
L
A
Risk
38
The Economics of Information

A Utility-Maximizing Model

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39
The basic model is shown in Figure 5-6 where
an individual is assumed to face two possible
outcomes (sometimes called states of the
world), but he or she does not know what
outcome will occur.
The person’s consumption in the two states is
denoted C1 and C2, and possible values are
recorded in the axes.
FIGURE 5-6: Utility Maximization
under Uncertainty
Certainty line
C2
C2E
E
B
C2A
D
U2
A
U1
CE1
40
CA1
C1
A Utility-Maximizing Model

At point A, the individual has considerably
more consumption in state 1 than in state 2.


41
This person might be willing to give up some
consumption in state 1 to consume more in state 2.
This might we accomplished by paying an insurance
premium in state 1 in order to increase consumption in
state 2 (when things go wrong).
A Utility-Maximizing Model

For example, if the terms at which
insurance can be bought are reflected in
the slope of the line AE, this person could
increase utility from U1 to U2 by
purchasing complete insurance and
moving to point E.

42
Buying complete insurance has allowed this
person to obtain CE1 (which equals CE2) with
certainty.
FIGURE 5-6: Utility Maximization
under Uncertainty
Certainty line
C2
C2E
E
B
C2A
D
U2
A
U1
C1E
43
C1A
C1
Balancing the Gains and Costs of
Information

Another way to improve his or her
situation would be to gather additional
information.

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44
Consumption will decrease in the good outcome but
increase in the bad outcome.
The issue is whether acquiring information will raise
utility above U1.
Point B, for example, represents a utility-improving
investment, but point D is a poor investment in
information.
Information Differences among
Economic Actors

The level of information that an individual
acquires will depend on how much the
information costs.

There are reasons to believe that information
costs may differ significantly among
individuals.

45
Sellers or large-scale repeat buyers of a good may
have greater access to information than first-time
buyers.