1. Choices Involving Risk
Chapter 11
Choices Involving Risk
McGraw-Hill/Irwin
Copyright © 2008 by The McGraw-Hill Companies, Inc. All Rights Reserved.

2. Main Topics What is a risk? Risk preference Insurance
Other methods of managing risk
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3. 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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4. 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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5. 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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6. Table 11.1
States of Nature, Probabilities, and Payoffs
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7. Table 11.2
Expected Value
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8. 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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9. 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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10. Figure 11.3: Consumption Bundles Example
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11. 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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12. Figure 11.4: Preferences for Risky Consumption Bundles
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13. 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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14. 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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15. Figure 11.6: Risk Aversion
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16. 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:
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17. 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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18. Figure 11.10: Expected Utility for a Risk-Averse Consumer
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19. 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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20. 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
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21. 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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22. Figure 11.12: Demand for Insurance
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23. Figure 11.13: Demand for Unfair Insurance
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24. 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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25. 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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26. 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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27. Figure 11.16: Hedging a Risky Venture
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28. 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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