Chapter 11
Choices Involving Risk
McGraw-Hill/Irwin
Copyright © 2008 by The McGraw-Hill Companies, Inc. All Rights Reserved.
Main Topics
What is a risk?
Risk preference
Insurance
Other methods of managing risk
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What is Risk?
Risk exists whenever the consequences of a
decision are uncertain
A state of nature is one possible way in which
events relevant to a risky decision can unfold
To analyze a risky decision, begin by
describing every state of nature
Once someone makes a choice, he
experiences only one state of nature
Can’t experience more than one state because each
state is described in a way that rules out the others
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Probability
Some states of nature are more likely than
others
Probability is a measure of the likelihood that
a given state will occur
A number between 0 and 1, or a percentage
Probability of 0 means a state is impossible,
probability of 1 means it’s certain
Add the probabilities of two states of nature to
obtain the probability that one of those two states
will occur
The probabilities of all states always add to 1; it’s
certain that something will happen
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Uncertain Payoffs
 Risky choices often have financial consequences,
payoffs
 Payoffs can be positive (gains) or negative (losses)
 The probability distribution of a set of payoffs gives
the likelihood that each possible payoff will occur
 To determine the average gain or loss from a risky
choice, can calculate its expected payoff
 Expected payoff of a risky financial choice is a
weighted average of all the possible payoffs, using the
probability of each payoff as its weight
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Table 11.1
States of Nature, Probabilities, and Payoffs
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Table 11.2
Expected Value
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Variability
Gauge financial risk by measuring the
variability of gains and losses
Generally, variability is low when range of likely
payoffs is narrow and high when range of likely
payoffs is wide
With little variability, the actual payoff is almost
always close to the expected payoff
Deviation is the difference between actual
payoff and expected payoff
Often, economists measure the variability of a
risky financial payoff by calculating variance or
standard deviation
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Risk Preferences
 Can think of a consumption bundle as a list of the
quantities of each good consumed in each possible
state of nature
 The guaranteed consumption line shows the
consumption bundles for which the level of
consumption does not depend on the state
 For bundles that do not lie on this line, the consumer’s
payoff is uncertain
 Can compute expected consumption for any particular bundle
 A constant expected consumption line shows all
risky consumption bundles with the same level of
expected consumption
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Figure 11.3: Consumption
Bundles Example
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Preferences and
Indifference Curves
If one bundle guarantees more of every good
than a second bundle, a consumer should
prefer the first
Reflects the More-Is-Better Principle
Does not have to guarantee a particular level of
consumption
Slope of an indifference curve indicates
willingness to shift consumption from one state
of nature to another
Depends on the probabilities of the states
Change in probabilities changes slopes of
indifference curves
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Figure 11.4: Preferences for Risky
Consumption Bundles
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Risk Aversion
A person is risk averse if, in comparing a
riskless bundle to a risky bundle with the same
level of expected consumption, he prefers the
riskless bundle
Risk averse individuals do not avoid risk at all costs
Usually willing to accept some risk provided they
receive adequate compensation in the form of
higher expected consumption
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Risk Premium
The certainty equivalent of a risky bundle is
the amount of consumption which, if provided
with certainty, would make the consumer
equally well off
For a risk-averse person, the certainty
equivalent of a risky bundle is always less than
expected consumption
Providing the same expected consumption with no
risk would make the individual better off
The risk premium of a risky bundle is the
difference between its expected consumption
and the consumer’s certainty equivalent
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Figure 11.6: Risk Aversion
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Expected Utility Functions
An expected utility function:
Assigns a benefit level to each possible state of
nature based only on what is consumed
Then takes the expected value of those benefits
It is a weighted average of all possible benefit
levels using the probability of each level as its
weight
Sample expected utility function:
U Fs , FH    W FS   1   W FH 
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Expected Utility and Risk Aversion
Can determine the consumer’s attitude toward
risk from the shaper of her benefit function,
W(F):
If W(F) is concave (flattens as F increases), she’s
risk averse
If it’s convex (gets steeper as F increases), she’s
risk loving
If it’s linear, she’s risk neutral
The greater the concavity, the greater the risk
aversion
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Figure 11.10: Expected Utility for
a Risk-Averse Consumer
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The Nature of Insurance
 People address a wide range of risks by purchasing
insurance policies
 An insurance policy is a contract that reduces the
financial loss associated with some risky event, such
as burglary
 The purchaser of an insurance policy is essentially
placing a bet
 Having paid M, the premium, the policy holder receives
B, the benefit, if a loss occurs, for a net gain of B – M
 If a loss doesn’t occur the consumer loses the premium
M
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Actuarial Fairness
 An insurance policy is actuarially fair if its expected
net payoff is zero
 Actuarial fairness requires:
   M   1   B  M   0
 So an actuarially fair insurance premium equals the
promised benefit times the probability of a loss
 Insurance policies are usually less than actuarially
fair because insurance companies must cover their
costs of operation
 On average purchasing such a policy reduces the
purchaser’s expected consumption
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Demand for Insurance
Risk-averse consumers are willing to purchase
insurance because it cancels out other risks
If the insurance is actuarially fair, a risk averse
consumer will purchase full insurance
With full insurance, the promised benefit equals the
potential loss
This does not depend on degree of risk aversion
If the insurance is less-than-actuarially fair, the
amount of insurance purchased depends on
degree of risk aversion
Risk neutral consumer will purchase none
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Figure 11.12: Demand for
Insurance
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Figure 11.13: Demand for
Unfair Insurance
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Other Methods of Managing Risk
 Four other strategies for managing risk
 Object of risk management is to make risky activities
more attractive by reducing the potential losses while
preserving much of the potential gains
 Risk sharing involves dividing a risky prospect among
several people
 Hedging is the practice of taking on two risky activities
with negatively correlated financial payoffs
 Diversification is the practice of undertaking many
risky activities each on a small scale
 People also often try to reduce risk through information
acquisition
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Figure 11.15: Risk Sharing
 Bundle A is initial bundle, and
riskless
 By investing, consumer can
move to B with higher
expected consumption
 Consumer prefers to avoid
risk associated with B
 With partners to split
investment and profits, can
reach points on the green line
 D is most preferred bundle
with risk sharing
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Hedging
Hedging is the practice of taking on two risky
activities with negatively correlated financial
payoffs
Two variables are negatively correlated if they tend
to move in the opposite direction
Bad news on one investment tends to be offset
by good news on the other
Insurance is a form of hedging
Benefit paid by a flood insurance policy is perfectly
negatively correlated with a loss from flooding
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Figure 11.16: Hedging a
Risky Venture
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Diversification
Diversification is the practice of undertaking
many risky activities, each on a small scale,
rather than a few risky activities on a large
scale
“Don’t put all your eggs in one basket”
Dividing investments among many activities reduces
risk
As correlation between the payoffs on the
investments increases, the risk-reducing effect
of diversification decreases
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