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Chapter 5 Choice Under Uncertainty

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Chapter 5Slide 2 Topics to be Discussed Describing Risk Preferences Toward Risk Reducing Risk The Demand for Risky Assets

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Chapter 5Slide 3 Introduction Choice with certainty is reasonably straightforward. How do we choose when certain variables such as income and prices are uncertain (i.e. making choices with risk)?

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Chapter 5Slide 4 Describing Risk To measure risk we must know: 1) All of the possible outcomes. 2) The likelihood that each outcome will occur (its probability).

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Chapter 5Slide 5 Describing Risk Interpreting Probability The likelihood that a given outcome will occur

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Chapter 5Slide 6 Describing Risk Interpreting Probability Objective Interpretation Based on the observed frequency of past events

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Chapter 5Slide 7 Describing Risk Interpreting Probability Subjective Based on perception or experience with or without an observed frequency Different information or different abilities to process the same information can influence the subjective probability

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Chapter 5Slide 8 Describing Risk Expected Value The weighted average of the payoffs or values resulting from all possible outcomes. The probabilities of each outcome are used as weights Expected value measures the central tendency; the payoff or value expected on average

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Chapter 5Slide 9 Describing Risk An Example Investment in offshore drilling exploration: Two outcomes are possible Success -- the stock price increase from $30 to $40/share Failure -- the stock price falls from $30 to $20/share

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Chapter 5Slide 10 Describing Risk An Example Objective Probability 100 explorations, 25 successes and 75 failures Probability (Pr) of success = 1/4 and the probability of failure = 3/4

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Chapter 5Slide 11 Describing Risk An Example: Expected Value (EV)

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Chapter 5Slide 12 Describing Risk Given: Two possible outcomes having payoffs X 1 and X 2 Probabilities of each outcome is given by Pr 1 & Pr 2

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Chapter 5Slide 13 Describing Risk Generally, expected value is written as:

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Chapter 5Slide 14 Describing Risk Variability The extent to which possible outcomes of an uncertain even may differ

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Chapter 5Slide 15 Describing Risk A Scenario Suppose you are choosing between two part-time sales jobs that have the same expected income ($1,500) The first job is based entirely on commission. The second is a salaried position. Variability

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Chapter 5Slide 16 Describing Risk A Scenario There are two equally likely outcomes in the first job--$2,000 for a good sales job and $1,000 for a modestly successful one. The second pays $1,510 most of the time (.99 probability), but you will earn $510 if the company goes out of business (.01 probability). Variability

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Chapter 5Slide 17 Income from Sales Jobs Job 1: Commission.52000.510001500 Job 2: Fixed salary.991510.015101500 Expected ProbabilityIncome ($)ProbabilityIncome ($)Income Outcome 1Outcome 2 Describing Risk

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Chapter 5Slide 18 Job 1 Expected Income Job 2 Expected Income Income from Sales Jobs Describing Risk

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Chapter 5Slide 19 While the expected values are the same, the variability is not. Greater variability from expected values signals greater risk. Deviation Difference between expected payoff and actual payoff Describing Risk

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Chapter 5Slide 20 Deviations from Expected Income ($) Job 1$2,000$500$1,000-$500 Job 21,51010510-900 Outcome 1 Deviation Outcome 2 Deviation Describing Risk

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Chapter 5Slide 21 Adjusting for negative numbers The standard deviation measures the square root of the average of the squares of the deviations of the payoffs associated with each outcome from their expected value. Variability Describing Risk

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Chapter 5Slide 22 Describing Risk The standard deviation is written: Variability

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Chapter 5Slide 23 Calculating Variance ($) Job 1$2,000$250,000$1,000 $250,000 $250,000 $500.00 Job 21,510100510 980,100 9,900 99.50 DeviationDeviation Deviation Standard Outcome 1 SquaredOutcome 2 Squared Squared Deviation Describing Risk

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Chapter 5Slide 24 Describing Risk The standard deviations of the two jobs are: *Greater Risk

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Chapter 5Slide 25 Describing Risk The standard deviation can be used when there are many outcomes instead of only two.

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Chapter 5Slide 26 Describing Risk Job 1 is a job in which the income ranges from $1000 to $2000 in increments of $100 that are all equally likely. Example

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Chapter 5Slide 27 Describing Risk Job 2 is a job in which the income ranges from $1300 to $1700 in increments of $100 that, also, are all equally likely. Example

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Chapter 5Slide 28 Outcome Probabilities for Two Jobs Income 0.1 $1000$1500$2000 0.2 Job 1 Job 2 Job 1 has greater spread: greater standard deviation and greater risk than Job 2. Probability

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Chapter 5Slide 29 Describing Risk Outcome Probabilities of Two Jobs (unequal probability of outcomes) Job 1: greater spread & standard deviation Peaked distribution: extreme payoffs are less likely

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Chapter 5Slide 30 Describing Risk Decision Making A risk avoider would choose Job 2: same expected income as Job 1 with less risk. Suppose we add $100 to each payoff in Job 1 which makes the expected payoff = $1600.

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Chapter 5Slide 31 Unequal Probability Outcomes Job 1 Job 2 The distribution of payoffs associated with Job 1 has a greater spread and standard deviation than those with Job 2. Income 0.1 $1000$1500$2000 0.2 Probability

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Chapter 5Slide 32 Income from Sales Jobs--Modified ($) Recall: The standard deviation is the square root of the deviation squared. Job 1$2,100$250,000$1,100$250,000$1,600$500 Job 21510100510980,1001,500 99.50 DeviationDeviationExpectedStandard Outcome 1 SquaredOutcome 2 SquaredIncomeDeviation

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Chapter 5Slide 33 Describing Risk Job 1: expected income $1,600 and a standard deviation of $500. Job 2: expected income of $1,500 and a standard deviation of $99.50 Which job? Greater value or less risk? Decision Making

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Chapter 5Slide 34 Suppose a city wants to deter people from double parking. The alternatives …... Describing Risk Example

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Chapter 5Slide 35 Assumptions: 1)Double-parking saves a person $5 in terms of time spent searching for a parking space. 2)The driver is risk neutral. 3)Cost of apprehension is zero. Example Describing Risk

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Chapter 5Slide 36 A fine of $5.01 would deter the driver from double parking. Benefit of double parking ($5) is less than the cost ($5.01) equals a net benefit that is less than 0. Example Describing Risk

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Chapter 5Slide 37 Increasing the fine can reduce enforcement cost: A $50 fine with a.1 probability of being caught results in an expected penalty of $5. A $500 fine with a.01 probability of being caught results in an expected penalty of $5. Example Describing Risk

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Chapter 5Slide 38 The more risk averse drivers are, the lower the fine needs to be in order to be effective. Example Describing Risk

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Chapter 5Slide 39 Preferences Toward Risk Choosing Among Risky Alternatives Assume Consumption of a single commodity The consumer knows all probabilities Payoffs measured in terms of utility Utility function given

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Chapter 5Slide 40 Preferences Toward Risk A person is earning $15,000 and receiving 13 units of utility from the job. She is considering a new, but risky job. Example

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Chapter 5Slide 41 Preferences Toward Risk She has a.50 chance of increasing her income to $30,000 and a.50 chance of decreasing her income to $10,000. She will evaluate the position by calculating the expected value (utility) of the resulting income. Example

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Chapter 5Slide 42 Preferences Toward Risk The expected utility of the new position is the sum of the utilities associated with all her possible incomes weighted by the probability that each income will occur. Example

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Chapter 5Slide 43 Preferences Toward Risk The expected utility can be written: E(u) = (1/2)u($10,000) + (1/2)u($30,000) = 0.5(10) + 0.5(18) = 14 E(u) of new job is 14 which is greater than the current utility of 13 and therefore preferred. Example

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Chapter 5Slide 44 Preferences Toward Risk Different Preferences Toward Risk People can be risk averse, risk neutral, or risk loving.

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Chapter 5Slide 45 Preferences Toward Risk Different Preferences Toward Risk Risk Averse: A person who prefers a certain given income to a risky income with the same expected value. A person is considered risk averse if they have a diminishing marginal utility of income The use of insurance demonstrates risk aversive behavior.

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Chapter 5Slide 46 Preferences Toward Risk A Scenario A person can have a $20,000 job with 100% probability and receive a utility level of 16. The person could have a job with a.5 chance of earning $30,000 and a.5 chance of earning $10,000. Risk Averse

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Chapter 5Slide 47 Preferences Toward Risk Expected Income = (0.5)($30,000) + (0.5)($10,000) = $20,000 Risk Averse

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Chapter 5Slide 48 Preferences Toward Risk Expected income from both jobs is the same -- risk averse may choose current job Risk Averse

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Chapter 5Slide 49 Preferences Toward Risk The expected utility from the new job is found: E(u) = (1/2)u ($10,000) + (1/2)u($30,000) E(u) = (0.5)(10) + (0.5)(18) = 14 E(u) of Job 1 is 16 which is greater than the E(u) of Job 2 which is 14. Risk Averse

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Chapter 5Slide 50 Preferences Toward Risk This individual would keep their present job since it provides them with more utility than the risky job. They are said to be risk averse. Risk Averse

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Chapter 5Slide 51 Income ($1,000) Utility The consumer is risk averse because she would prefer a certain income of $20,000 to a gamble with a.5 probability of $10,000 and a.5 probability of $30,000. E 10 1520 13 14 16 18 0 1630 A B C D Risk Averse Preferences Toward Risk

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Chapter 5Slide 52 Preferences Toward Risk A person is said to be risk neutral if they show no preference between a certain income, and an uncertain one with the same expected value. Risk Neutral

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Chapter 5Slide 53 Income ($1,000) 1020 Utility 0 30 6 A E C 12 18 The consumer is risk neutral and is indifferent between certain events and uncertain events with the same expected income. Preferences Toward Risk Risk Neutral

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Chapter 5Slide 54 Preferences Toward Risk A person is said to be risk loving if they show a preference toward an uncertain income over a certain income with the same expected value. Examples: Gambling, some criminal activity Risk Loving

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Chapter 5Slide 55 Income ($1,000) Utility 0 3 102030 A E C 8 18 The consumer is risk loving because she would prefer the gamble to a certain income. Preferences Toward Risk Risk Loving

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Chapter 5Slide 56 Preferences Toward Risk The risk premium is the amount of money that a risk-averse person would pay to avoid taking a risk. Risk Premium

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Chapter 5Slide 57 Preferences Toward Risk A Scenario The person has a.5 probability of earning $30,000 and a.5 probability of earning $10,000 (expected income = $20,000). The expected utility of these two outcomes can be found: E(u) =.5(18) +.5(10) = 14 Risk Premium

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Chapter 5Slide 58 Preferences Toward Risk Question How much would the person pay to avoid risk? Risk Premium

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Chapter 5Slide 59 Income ($1,000) Utility 0 1016 Here, the risk premium is $4,000 because a certain income of $16,000 gives the person the same expected utility as the uncertain income that has an expected value of $20,000. 10 18 3040 20 14 A C E G 20 F Risk Premium Preferences Toward Risk Risk Premium

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Chapter 5Slide 60 Preferences Toward Risk Variability in potential payoffs increase the risk premium. Example: A job has a.5 probability of paying $40,000 (utility of 20) and a.5 chance of paying 0 (utility of 0). Risk Aversion and Income

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Chapter 5Slide 61 Preferences Toward Risk Example: The expected income is still $20,000, but the expected utility falls to 10. Expected utility =.5u($) +.5u($40,000) = 0 +.5(20) = 10 Risk Aversion and Income

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Chapter 5Slide 62 Preferences Toward Risk Example: The certain income of $20,000 has a utility of 16. If the person is required to take the new position, their utility will fall by 6. Risk Aversion and Income

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Chapter 5Slide 63 Preferences Toward Risk Example: The risk premium is $10,000 (i.e. they would be willing to give up $10,000 of the $20,000 and have the same E(u) as the risky job. Risk Aversion and Income

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Chapter 5Slide 64 Preferences Toward Risk Therefore, it can be said that the greater the variability, the greater the risk premium. Risk Aversion and Income

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Chapter 5Slide 65 Preferences Toward Risk Combinations of expected income & standard deviation of income that yield the same utility Indifference Curve

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Chapter 5Slide 66 Risk Aversion and Indifference Curves Standard Deviation of Income Expected Income Highly Risk Averse:An increase in standard deviation requires a large increase in income to maintain satisfaction. U1U1 U2U2 U3U3

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Chapter 5Slide 67 Risk Aversion and Indifference Curves Standard Deviation of Income Expected Income Slightly Risk Averse: A large increase in standard deviation requires only a small increase in income to maintain satisfaction. U1U1 U2U2 U3U3

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Chapter 5Slide 68 Business Executives and the Choice of Risk Study of 464 executives found that: 20% were risk neutral 40% were risk takers 20% were risk adverse 20% did not respond Example

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Chapter 5Slide 69 Those who liked risky situations did so when losses were involved. When risks involved gains the same, executives opted for less risky situations. Example Business Executives and the Choice of Risk

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Chapter 5Slide 70 The executives made substantial efforts to reduce or eliminate risk by delaying decisions and collecting more information. Example Business Executives and the Choice of Risk

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Chapter 5Slide 71 Reducing Risk Three ways consumers attempt to reduce risk are: 1) Diversification 2) Insurance 3) Obtaining more information

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Chapter 5Slide 72 Reducing Risk Diversification Suppose a firm has a choice of selling air conditioners, heaters, or both. The probability of it being hot or cold is 0.5. The firm would probably be better off by diversification.

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Chapter 5Slide 73 Income from Sales of Appliances Air conditioner sales$30,000$12,000 Heater sales12,00030,000 * 0.5 probability of hot or cold weather Hot Weather Cold Weather

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Chapter 5Slide 74 Reducing Risk If the firms sells only heaters or air conditioners their income will be either $12,000 or $30,000. Their expected income would be: 1/2($12,000) + 1/2($30,000) = $21,000 Diversification

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Chapter 5Slide 75 Reducing Risk If the firm divides their time evenly between appliances their air conditioning and heating sales would be half their original values. Diversification

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Chapter 5Slide 76 Reducing Risk If it were hot, their expected income would be $15,000 from air conditioners and $6,000 from heaters, or $21,000. If it were cold, their expected income would be $6,000 from air conditioners and $15,000 from heaters, or $21,000. Diversification

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Chapter 5Slide 77 Reducing Risk With diversification, expected income is $21,000 with no risk. Diversification

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Chapter 5Slide 78 Reducing Risk Firms can reduce risk by diversifying among a variety of activities that are not closely related. Diversification

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Chapter 5Slide 79 Reducing Risk Discussion Questions How can diversification reduce the risk of investing in the stock market? Can diversification eliminate the risk of investing in the stock market? The Stock Market

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Chapter 5Slide 80 Reducing Risk Risk averse are willing to pay to avoid risk. If the cost of insurance equals the expected loss, risk averse people will buy enough insurance to recover fully from a potential financial loss. Insurance

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Chapter 5Slide 81 The Decision to Insure No$40,000$50,000$49,000$9,055 Yes49,00049,00049,0000 InsuranceBurglary No Burglary Expected Standard (Pr =.1)(Pr =.9) Wealth Deviation

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Chapter 5Slide 82 Reducing Risk While the expected wealth is the same, the expected utility with insurance is greater because the marginal utility in the event of the loss is greater than if no loss occurs. Purchases of insurance transfers wealth and increases expected utility. Insurance

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Chapter 5Slide 83 Reducing Risk Although single events are random and largely unpredictable, the average outcome of many similar events can be predicted. The Law of Large Numbers

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Chapter 5Slide 84 Reducing Risk Examples A single coin toss vs. large number of coins Whom will have a car wreck vs. the number of wrecks for a large group of drivers The Law of Large Numbers

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Chapter 5Slide 85 Reducing Risk Assume: 10% chance of a $10,000 loss from a home burglary Expected loss =.10 x $10,000 = $1,000 with a high risk (10% chance of a $10,000 loss) 100 people face the same risk Actuarial Fairness

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Chapter 5Slide 86 Reducing Risk Then: $1,000 premium generates a $100,000 fund to cover losses Actual Fairness When the insurance premium = expected payout Actuarial Fairness

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Chapter 5Slide 87 The Value of Title Insurance When Buying a House A Scenario: Price of a house is $200,000 5% chance that the seller does not own the house Example

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Chapter 5Slide 88 The Value of Title Insurance When Buying a House Risk neutral buyer would pay: Example

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Chapter 5Slide 89 The Value of Title Insurance When Buying a House Risk averse buyer would pay much less By reducing risk, title insurance increases the value of the house by an amount far greater than the premium. Example

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Chapter 5Slide 90 Reducing Risk Value of Complete Information The difference between the expected value of a choice with complete information and the expected value when information is incomplete. The Value of Information

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Chapter 5Slide 91 Reducing Risk Suppose a store manager must determine how many fall suits to order: 100 suits cost $180/suit 50 suits cost $200/suit The price of the suits is $300 The Value of Information

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Chapter 5Slide 92 Reducing Risk Suppose a store manager must determine how many fall suits to order: Unsold suits can be returned for half cost. The probability of selling each quantity is.50. The Value of Information

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Chapter 5Slide 93 The Decision to Insure 1. Buy 50 suits$5,000$5,000$5,000 2. Buy 100 suits1,50012,0006,750 Expected Sale of 50Sale of 100 Profit

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Chapter 5Slide 94 Reducing Risk With incomplete information: Risk Neutral: Buy 100 suits Risk Averse: Buy 50 suits

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Chapter 5Slide 95 Reducing Risk The expected value with complete information is $8,500. 8,500 =.5(5,000) +.5(12,000) The expected value with uncertainty (buy 100 suits) is $6,750. The Value of Information

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Chapter 5Slide 96 Reducing Risk The value of complete information is $1,750, or the difference between the two (the amount the store owner would be willing to pay for a marketing study). The Value of Information

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Chapter 5Slide 97 Per capita milk consumption has fallen over the years The milk producers engaged in market research to develop new sales strategies to encourage the consumption of milk. Reducing Risk The Value of Information: Example

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Chapter 5Slide 98 Findings Milk demand is seasonal with the greatest demand in the spring E p is negative and small E I is positive and large Reducing Risk The Value of Information: Example

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Chapter 5Slide 99 Milk advertising increases sales most in the spring. Allocating advertising based on this information in New York increased sales by $4,046,557 and profits by 9%. The cost of the information was relatively low, while the value was substantial. Reducing Risk The Value of Information: Example

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Chapter 5Slide 100 Assets Something that provides a flow of money or services to its owner. The flow of money or services can be explicit (dividends) or implicit (capital gain). The Demand for Risky Assets

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Chapter 5Slide 101 Capital Gain An increase in the value of an asset, while a decrease is a capital loss. The Demand for Risky Assets

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Chapter 5Slide 102 The Demand for Risky Assets Risky Asset Provides an uncertain flow of money or services to its owner. Examples apartment rent, capital gains, corporate bonds, stock prices Risky & Riskless Assets

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Chapter 5Slide 103 The Demand for Risky Assets Riskless Asset Provides a flow of money or services that is known with certainty. Examples short-term government bonds, short- term certificates of deposit Risky & Riskless Assets

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Chapter 5Slide 104 The Demand for Risky Assets Asset Returns Return on an Asset The total monetary flow of an asset as a fraction of its price. Real Return of an Asset The simple (or nominal) return less the rate of inflation.

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Chapter 5Slide 105 The Demand for Risky Assets Asset Returns

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Chapter 5Slide 106 The Demand for Risky Assets Expected Return Return that an asset should earn on average Expected vs. Actual Returns

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Chapter 5Slide 107 The Demand for Risky Assets Actual Return Return that an asset earns Expected vs. Actual Returns

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Chapter 5Slide 108 Investments--Risk and Return (1926-1999) Common stocks (S&P 500)9.520.2 Long-term corporate bonds2.78.3 U.S. Treasury bills0.63.2 Risk Real Rate of(standard Return (%)deviation,%)

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Chapter 5Slide 109 The Demand for Risky Assets Higher returns are associated with greater risk. The risk-averse investor must balance risk relative to return Expected vs. Actual Returns

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Chapter 5Slide 110 The Demand for Risky Assets An investor is choosing between T-Bills and stocks: T-bills (riskless) versus Stocks (risky) R f = the return on risk free T-bills Expected return equals actual return when there is no risk The Trade-Off Between Risk and Return

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Chapter 5Slide 111 The Demand for Risky Assets An investor is choosing between T-Bills and stocks: R m = the expected return on stocks r m = the actual returns on stock The Trade-Off Between Risk and Return

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Chapter 5Slide 112 The Demand for Risky Assets At the time of the investment decision, we know the set of possible outcomes and the likelihood of each, but we do not know what particular outcome will occur. The Trade-Off Between Risk and Return

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Chapter 5Slide 113 The Demand for Risky Assets The risky asset will have a higher expected return than the risk free asset (R m > R f ). Otherwise, risk-averse investors would buy only T-bills. The Trade-Off Between Risk and Return

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Chapter 5Slide 114 The Demand for Risky Assets How to allocate savings: b = fraction of savings in the stock market 1 - b = fraction in T-bills The Investment Portfolio

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Chapter 5Slide 115 The Demand for Risky Assets Expected Return: R p : weighted average of the expected return on the two assets R p = bR m + (1-b)R f The Investment Portfolio

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Chapter 5Slide 116 The Demand for Risky Assets Expected Return: If R m = 12%, R f = 4%, and b = 1/2 R p = 1/2(.12) + 1/2(.04) = 8% The Investment Portfolio

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Chapter 5Slide 117 The Demand for Risky Assets Question How risky is their portfolio? The Investment Portfolio

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Chapter 5Slide 118 The Demand for Risky Assets Risk (standard deviation) of the portfolio is the fraction of the portfolio invested in the risky asset times the standard deviation of that asset: The Investment Portfolio

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Chapter 5Slide 119 The Demand for Risky Assets Determining b: The Investors Choice Problem

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Chapter 5Slide 120 The Demand for Risky Assets Determining b: The Investors Choice Problem

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Chapter 5Slide 121 The Demand for Risky Assets Observations 1)The final equation is a budget line describing the trade- off between risk and expected return. Risk and the Budget Line

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Chapter 5Slide 122 The Demand for Risky Assets Observations: 2)Is an equation for a straight line: 3) Risk and the Budget Line

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Chapter 5Slide 123 The Demand for Risky Assets Observations 3)Expected return, R P, increases as risk increases. 4)The slope is the price of risk or the risk-return trade-off. Risk and the Budget Line

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Chapter 5Slide 124 Choosing Between Risk and Return 0 Expected Return,R p U 2 is the optimal choice of those obtainable, since it gives the highest return for a given risk and is tangent to the budget line. RfRf Budget Line RmRm R*R* U2U2 U1U1 U3U3

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Chapter 5Slide 125 RfRf Budget line The Choices of Two Different Investors 0 Expected Return,R p Given the same budget line, investor A chooses low return-low risk, while investor B chooses high return- high risk. UAUA RARA UBUB RBRB RmRm

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Chapter 5Slide 126 RfRf Budget line Buying Stocks on Margin 0 Expected Return,R p UAUA RARA U A : High risk aversion --Stock & T-bill portfolio UBUB RBRB RmRm U A : Low risk aversion --The investor would invest more than 100% of their wealth by borrowing or buying on the margin.

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Chapter 5Slide 127 Investing in the Stock Market Observations Percent of American families who had directly or indirectly invested in the stock market 1989 = 32% 1995 = 41%

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Chapter 5Slide 128 Investing in the Stock Market Observations Share of wealth in the stock market 1989 = 26% 1995 = 40%

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Chapter 5Slide 129 Investing in the Stock Market Observations Participation in the stock market by age Less than 35 1989 = 23% 1995 = 29% More than 35 Small increase

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Chapter 5Slide 130 Investing in the Stock Market What Do You Think? Why are more people investing in the stock market?

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Chapter 5Slide 131 Summary Consumers and managers frequently make decisions in which there is uncertainty about the future. Consumers and investors are concerned about the expected value and the variability of uncertain outcomes.

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Chapter 5Slide 132 Summary Facing uncertain choices, consumers maximize their expected utility, and average of the utility associated with each outcome, with the associated probabilities serving as weights. A person may be risk averse, risk neutral or risk loving.

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Chapter 5Slide 133 Summary The maximum amount of money that a risk-averse person would pay to avoid risk is the risk premium. Risk can be reduced by diversification, purchasing insurance, and obtaining additional information.

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Chapter 5Slide 134 Summary The law of large numbers enables insurance companies to provide actuarially fair insurance for which the premium paid equals the expected value of the loss being insured against. Consumer theory can be applied to decisions to invest in risky assets.

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End of Chapter 5 Choice Under Uncertainty

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