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Part 3Part 3
Uncertainty and Strategy
© 2006 Thomson Learning/South-Western
Chapter 5Chapter 5
Uncertainty
© 2006 Thomson Learning/South-Western
3
Probability
Probability of an event happening; relative frequency with which an event occurs 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.
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Expected Value
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 = -$1) if a head comes up, the expected value of game for both players is
.0)1$(2
1)1$(
2
1
2
1
2
121 XX
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Expected Value If game changes so that, from Smith’s point of
view, X1 = $10, and X2 = -$1, expected value for Smith would be:
Because Smith would stand to win $4.50 on average, she might be willing to pay Jones up to this amount to play.
Fair games are games that cost precisely their expected value.
.50.4$)1$(2
1)10($
2
1
2
1
2
121 XX
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Risk Aversion
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.
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Diminishing Marginal Utility
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.
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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. Gain in utility due to increased income from
$1000 to $2000 exceeds gain from $49,000 to $50,000.
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Graphical Analysis of Risk Aversion
Figure 5-1 shows person with three options. Contender may: 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.
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U
Income(thousandsof dollars)
0 35 40 503020
Utility
FIGURE 5-1: Risk Aversion
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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). Average utility is U2 < U3.
The utility (U1 < U2) of the $15,000 bet is average of utility of winning ($50,000) and losing ($20,000).
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U
U3U2
U1
Income(thousandsof dollars)
0 35 40 503020
Utility
FIGURE 5-1: Risk Aversion
33
13
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 risk-free incomes to risky incomes that offer less utility. 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.
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Methods of Reducing Risk: Insurance
Figure 5-2 shows motive for buying insurance.
Assume that during next year person with $35,000 current income faces 50 percent chance of incurring $15,000 in unexpected medical bills.
Without insurance, person’s utility would be U1--utility of average of $20,000 and $35,000.
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U
U1
Income(thousandsof dollars)
0 25 3520
Utility
FIGURE 5-2: Insurance Reduces Risk
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Fair Insurance
Fair insurance: premium equals expected value of loss. 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
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UU2
U1
Income(thousandsof dollars)
0 27.525 3520
Utility
FIGURE 15.2: Insurance Reduces Risk
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Unfair Insurance
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 U0.
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UU2
U1
U0
Income(thousandsof dollars)
0 27.52523 3520
Utility
FIGURE 5-2: Insurance Reduces Risk
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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.
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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. Suitably spreading risk may increase utility
above that obtain by a single transaction.
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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 Each company’s stock costs $1, but value will
increase to $2 if company does well during next year.
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Diversification
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.
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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. 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.
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U
U1
Income(thousandsof dollars)
0 35 5020
Utility
FIGURE 5-3: Diversification Reduces Risk
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TABLE 5-1: Possible Outcomes from Investing in Two Companies
Company B’s Performance Poor Good
Poor $20,000 $35,000 Company A’s Performance Good 35,000 50,000
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Diversification
Each of four outcomes is equally likely; with half of cases, investor ends up with original $35,000. 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.
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UD
E
C
U2
U1
Income(thousandsof dollars)
0 35 5020
Utility
FIGURE 5-3: Diversification Reduces Risk
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Attributes of Options
Specification of underlying transaction
Definition of period during which option may be exercised
Price of option
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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 Expected value of transaction Variability of value of transaction.
31
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.
32
Investors’ Market Options 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.
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Annualreturn
Risk
A
C
Market Line
B
Figure 5-4: Market Options for Investors
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Investors’ Market Options 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.
35
Choices by Individual Investors
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.
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Choices by Individual Investors
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).
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Choices by Individual Investors
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.
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Annualreturn
Risk
A
L
Market Line
Figure 5-5: Choices by Individual Investors
M
NUI
UII
UIII
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The Economics of Information
A Utility-Maximizing Model 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.
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U1
U2
D
B
A
E
Certainty line
C2E
C2A
C1CE1 CA
1
C2
FIGURE 5-6: Utility Maximization under Uncertainty
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A Utility-Maximizing Model
At point A, the individual has considerably more consumption in state 1 than in state 2. This person might be willing to give up some
consumption in state 1 to consume more in state 2. This might be accomplished by paying an insurance
premium in state 1 in order to increase consumption in state 2 (when things go wrong).
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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. Buying complete insurance has allowed this
person to obtain CE1 (which equals CE
2) with certainty.
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U1
U2
D
B
A
E
Certainty line
C2E
C2A
C1C1E C1
A
C2
FIGURE 5-6: Utility Maximization under Uncertainty
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Balancing the Gains and Costs of Information
Another way to improve his or her situation would be to gather additional information. 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.
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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.
Sellers or large-scale repeat buyers of a good may have greater access to information than first-time buyers.
