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 preferenceInsuranceOther 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 natureCan’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 occurA number between 0 and 1, or a percentageProbability of 0 means a state is impossible,

probability of 1 means it’s certainAdd 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 11.1States of Nature, Probabilities, and Payoffs

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Table 11.2Expected Value

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Variability

Gauge financial risk by measuring the variability of gains and lossesGenerally, 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 andIndifference Curves

If one bundle guarantees more of every good than a second bundle, a consumer should prefer the firstReflects the More-Is-Better PrincipleDoes not have to guarantee a particular level of

consumptionSlope of an indifference curve indicates

willingness to shift consumption from one state of nature to anotherDepends on the probabilities of the statesChange 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 bundleRisk averse individuals do not avoid risk at all costsUsually 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 consumptionProviding the same expected consumption with no

risk would make the individual better offThe 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 consumedThen takes the expected value of those benefitsIt is a weighted average of all possible benefit

levels using the probability of each level as its weight

Sample expected utility function: HSHs FWFWFFU 1,

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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 averseIf it’s convex (gets steeper as F increases), she’s

risk lovingIf 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:

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

01 MBM

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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 insuranceWith full insurance, the promised benefit equals the

potential lossThis 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 aversionRisk neutral consumer will purchase none

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Figure 11.12: Demand for Insurance

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Figure 11.13: Demand forUnfair 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 payoffsTwo variables are negatively correlated if they tend

to move in the opposite directionBad news on one investment tends to be offset

by good news on the otherInsurance 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 aRisky 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

riskAs correlation between the payoffs on the

investments increases, the risk-reducing effect of diversification decreases

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