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**Choices Involving Risk**

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

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**Main Topics What is a risk? Risk preference Insurance**

Other methods of managing risk 11-2

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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 11-3

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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 11-4

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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 11-5

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Table 11.1 States of Nature, Probabilities, and Payoffs 11-6

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Table 11.2 Expected Value 11-7

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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 11-8

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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 11-9

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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 11-11

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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 11-13

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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 11-14

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Figure 11.6: Risk Aversion 11-15

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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: 11-16

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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 11-17

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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 11-19

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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 11-20

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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 11-21

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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 11-24

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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 11-25

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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 11-26

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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 11-28

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