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1、Chapter 5Choice Under Choice Under UncertaintyUncertainty1Chapter 1Topics to be DiscussednDescribing RisknPreferences Toward RisknReducing RisknThe Demand for Risky Assets2Chapter 1IntroductionnChoice with certainty is reasonably straightforward.nHow do we choose when certain variables such as incom
2、e and prices are uncertain(i.e.making choices with risk)?3Chapter 1Describing RisknTo measure risk we must know:1)All of the possible outcomes.2)The likelihood that each outcome will occur(its probability).4Chapter 1Describing RisknInterpreting ProbabilitylThe likelihood that a given outcome will oc
3、cur5Chapter 1Describing RisknInterpreting ProbabilitylObjective InterpretationuBased on the observed frequency of past events6Chapter 1Describing RisknInterpreting ProbabilitylSubjectiveuBased on perception or experience with or without an observed frequencylDifferent information or different abilit
4、ies to process the same information can influence the subjective probability7Chapter 1Describing RisknExpected ValuelThe weighted average of the payoffs or values resulting from all possible outcomes.uThe probabilities of each outcome are used as weightsuExpected value measures the central tendency;
5、the payoff or value expected on average8Chapter 1Describing RisknAn ExamplelInvestment in offshore drilling exploration:lTwo outcomes are possibleuSuccess-the stock price increase from$30 to$40/shareuFailure-the stock price falls from$30 to$20/share9Chapter 1Describing RisknAn ExamplelObjective Prob
6、abilityu100 explorations,25 successes and 75 failuresuProbability(Pr)of success=1/4 and the probability of failure=3/410Chapter 1Describing RisknAn Example:Expected Value(EV)11Chapter 1Describing RisknGiven:lTwo possible outcomes having payoffs X1 and X2lProbabilities of each outcome is given by Pr1
7、&Pr212Chapter 1Describing RisknGenerally,expected value is written as:13Chapter 1Describing RisknVariabilitylThe extent to which possible outcomes of an uncertain even may differ14Chapter 1Describing RisknA ScenariolSuppose you are choosing between two part-time sales jobs that have the same expecte
8、d income($1,500)lThe first job is based entirely on commission.lThe second is a salaried position.Variability15Chapter 1Describing RisknA ScenariolThere are two equally likely outcomes in the first job-$2,000 for a good sales job and$1,000 for a modestly successful one.lThe second pays$1,510 most of
9、 the time(.99 probability),but you will earn$510 if the company goes out of business(.01 probability).Variability16Chapter 1Income from Sales JobsJob 1:Commission.52000.510001500Job 2:Fixed salary.991510.015101500ExpectedProbabilityIncome($)ProbabilityIncome($)IncomeOutcome 1Outcome 2Describing Risk
10、17Chapter 1nJob 1 Expected IncomenJob 2 Expected IncomeIncome from Sales JobsDescribing Risk18Chapter 1nWhile the expected values are the same,the variability is not.nGreater variability from expected values signals greater risk.nDeviationlDifference between expected payoff and actual payoffDescribi
11、ng Risk19Chapter 1Deviations from Expected Income($)Job 1$2,000$500$1,000-$500Job 21,51010510-900 Outcome 1 Deviation Outcome 2 DeviationDescribing Risk20Chapter 1nAdjusting for negative numbersnThe standard deviation measures the square root of the average of the squares of the deviations of the pa
12、yoffs associated with each outcome from their expected value.VariabilityDescribing Risk21Chapter 1Describing RisknThe standard deviation is written:Variability22Chapter 1Calculating Variance($)Job 1$2,000$250,000$1,000$250,000$250,000$500.00Job 21,510100510 980,100 9,900 99.50DeviationDeviation Devi
13、ation Standard Outcome 1 SquaredOutcome 2 Squared Squared DeviationDescribing Risk23Chapter 1Describing RisknThe standard deviations of the two jobs are:*Greater Risk24Chapter 1Describing RisknThe standard deviation can be used when there are many outcomes instead of only two.25Chapter 1Describing R
14、isknJob 1 is a job in which the income ranges from$1000 to$2000 in increments of$100 that are all equally likely.Example26Chapter 1Describing RisknJob 2 is a job in which the income ranges from$1300 to$1700 in increments of$100 that,also,are all equally likely.Example27Chapter 1Outcome Probabilities
15、 for Two JobsIncome0.1$1000$1500$20000.2Job 1Job 2Job 1 has greater spread:greaterstandard deviationand greater riskthan Job 2.Probability28Chapter 1Describing RisknOutcome Probabilities of Two Jobs(unequal probability of outcomes)lJob 1:greater spread&standard deviationlPeaked distribution:extreme
16、payoffs are less likely29Chapter 1Describing RisknDecision MakinglA risk avoider would choose Job 2:same expected income as Job 1 with less risk.lSuppose we add$100 to each payoff in Job 1 which makes the expected payoff=$1600.30Chapter 1Unequal Probability OutcomesJob 1Job 2The distribution of payo
17、ffsassociated with Job 1 has a greater spread and standarddeviation than those with Job 2.Income0.1$1000$1500$20000.2Probability31Chapter 1Income from Sales Jobs-Modified($)Recall:The standard deviation is the square root of the deviation squared.Job 1$2,100$250,000$1,100$250,000$1,600$500Job 215101
18、00510980,1001,500 99.50DeviationDeviationExpectedStandard Outcome 1 SquaredOutcome 2 SquaredIncomeDeviation32Chapter 1Describing RisknJob 1:expected income$1,600 and a standard deviation of$500.nJob 2:expected income of$1,500 and a standard deviation of$99.50nWhich job?lGreater value or less risk?De
19、cision Making33Chapter 1nSuppose a city wants to deter people from double parking.nThe alternatives.Describing RiskExample34Chapter 1nAssumptions: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
20、Describing Risk35Chapter 1nA fine of$5.01 would deter the driver from double parking.lBenefit of double parking($5)is less than the cost($5.01)equals a net benefit that is less than 0.ExampleDescribing Risk36Chapter 1nIncreasing the fine can reduce enforcement cost:lA$50 fine with a.1 probability of
21、 being caught results in an expected penalty of$5.lA$500 fine with a.01 probability of being caught results in an expected penalty of$5.ExampleDescribing Risk37Chapter 1nThe more risk averse drivers are,the lower the fine needs to be in order to be effective.ExampleDescribing Risk38Chapter 1Preferen
22、ces Toward RisknChoosing Among Risky AlternativeslAssumeuConsumption of a single commodityuThe consumer knows all probabilitiesuPayoffs measured in terms of utilityuUtility function given39Chapter 1Preferences Toward RisknA person is earning$15,000 and receiving 13 units of utility from the job.nShe
23、 is considering a new,but risky job.Example40Chapter 1Preferences Toward RisknShe has a.50 chance of increasing her income to$30,000 and a.50 chance of decreasing her income to$10,000.nShe will evaluate the position by calculating the expected value(utility)of the resulting income.Example41Chapter 1
24、Preferences Toward RisknThe 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.Example42Chapter 1Preferences Toward RisknThe expected utility can be written:lE(u)=(1/2)u($10,000)+(1/2)u($30
25、,000)=0.5(10)+0.5(18)=14lE(u)of new job is 14 which is greater than the current utility of 13 and therefore preferred.Example43Chapter 1Preferences Toward RisknDifferent Preferences Toward RisklPeople can be risk averse,risk neutral,or risk loving.44Chapter 1Preferences Toward RisknDifferent Prefere
26、nces Toward RisklRisk Averse:A person who prefers a certain given income to a risky income with the same expected value.lA person is considered risk averse if they have a diminishing marginal utility of incomeuThe use of insurance demonstrates risk aversive behavior.45Chapter 1Preferences Toward Ris
27、knA ScenariolA person can have a$20,000 job with 100%probability and receive a utility level of 16.lThe person could have a job with a.5 chance of earning$30,000 and a.5 chance of earning$10,000.Risk Averse46Chapter 1Preferences Toward RisknExpected Income=(0.5)($30,000)+(0.5)($10,000)=$20,000Risk A
28、verse47Chapter 1Preferences Toward RisknExpected income from both jobs is the same-risk averse may choose current jobRisk Averse48Chapter 1Preferences Toward RisknThe expected utility from the new job is found:lE(u)=(1/2)u($10,000)+(1/2)u($30,000)lE(u)=(0.5)(10)+(0.5)(18)=14uE(u)of Job 1 is 16 which
29、 is greater than the E(u)of Job 2 which is 14.Risk Averse49Chapter 1Preferences Toward RisknThis individual would keep their present job since it provides them with more utility than the risky job.nThey are said to be risk averse.Risk Averse50Chapter 1Income($1,000)UtilityThe consumer is riskaverse
30、because shewould prefer a certainincome of$20,000 to agamble with a.5 probabilityof$10,000 and a.5probability of$30,000.E101015201314161801630ABCDRisk AversePreferences Toward Risk51Chapter 1Preferences Toward RisknA person is said to be risk neutral if they show no preference between a certain inco
31、me,and an uncertain one with the same expected value.Risk Neutral52Chapter 1Income($1,000)1020Utility0306AEC1218The consumer is riskneutral and is indifferentbetween certain eventsand uncertain eventswith the same expected income.Preferences Toward RiskRisk Neutral53Chapter 1Preferences Toward Riskn
32、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.lExamples:Gambling,some criminal activityRisk Loving54Chapter 1Income($1,000)Utility03102030AEC818The consumer is riskloving because shewould prefer the gamble t
33、o a certain income.Preferences Toward RiskRisk Loving55Chapter 1Preferences Toward RisknThe risk premium is the amount of money that a risk-averse person would pay to avoid taking a risk.Risk Premium56Chapter 1Preferences Toward RisknA ScenariolThe person has a.5 probability of earning$30,000 and a.
34、5 probability of earning$10,000(expected income=$20,000).lThe expected utility of these two outcomes can be found:uE(u)=.5(18)+.5(10)=14Risk Premium57Chapter 1Preferences Toward RisknQuestionlHow much would the person pay to avoid risk?Risk Premium58Chapter 1Income($1,000)Utility01016Here,the risk p
35、remiumis$4,000 because a certain income of$16,000gives the person the sameexpected utility as the uncertain income thathas an expected value of$20,000.101830402014ACEG20FRisk PremiumPreferences Toward RiskRisk Premium59Chapter 1Preferences Toward RisknVariability in potential payoffs increase the ri
36、sk premium.nExample:lA 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 Income60Chapter 1Preferences Toward RisknExample:lThe expected income is still$20,000,but the expected utility falls to 10.lExpected utility=.5u($)+.5u($40,000)=0+
37、.5(20)=10Risk Aversion and Income61Chapter 1Preferences Toward RisknExample:lThe certain income of$20,000 has a utility of 16.lIf the person is required to take the new position,their utility will fall by 6.Risk Aversion and Income62Chapter 1Preferences Toward RisknExample:lThe risk premium is$10,00
38、0(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 Income63Chapter 1Preferences Toward RisknTherefore,it can be said that the greater the variability,the greater the risk premium.Risk Aversion and Income64Chapter 1Preferences Toward R
39、isknCombinations of expected income&standard deviation of income that yield the same utilityIndifference Curve65Chapter 1Risk Aversion andIndifference CurvesStandard Deviation of Income ExpectedIncomeHighly Risk Averse:Anincrease in standarddeviation requires a large increase in income to maintainsa
40、tisfaction.U1U2U366Chapter 1Risk Aversion andIndifference CurvesStandard Deviation of Income ExpectedIncomeSlightly Risk Averse:A large increase in standarddeviation requires only a small increase in incometo maintain satisfaction.U1U2U367Chapter 1Business Executivesand the Choice of RisknStudy of 4
41、64 executives found that:l20%were risk neutrall40%were risk takersl20%were risk adversel20%did not respondExample68Chapter 1nThose who liked risky situations did so when losses were involved.nWhen risks involved gains the same,executives opted for less risky situations.ExampleBusiness Executivesand
42、the Choice of Risk69Chapter 1nThe executives made substantial efforts to reduce or eliminate risk by delaying decisions and collecting more information.ExampleBusiness Executivesand the Choice of Risk70Chapter 1Reducing RisknThree ways consumers attempt to reduce risk are:1)Diversification2)Insuranc
43、e3)Obtaining more information71Chapter 1Reducing RisknDiversification lSuppose a firm has a choice of selling air conditioners,heaters,or both.lThe probability of it being hot or cold is 0.5.lThe firm would probably be better off by diversification.72Chapter 1Income from Sales of AppliancesAir condi
44、tioner sales$30,000$12,000Heater sales12,00030,000 *0.5 probability of hot or cold weatherHot Weather Cold Weather73Chapter 1Reducing RisknIf the firms sells only heaters or air conditioners their income will be either$12,000 or$30,000.nTheir expected income would be:l1/2($12,000)+1/2($30,000)=$21,0
45、00Diversification74Chapter 1Reducing RisknIf the firm divides their time evenly between appliances their air conditioning and heating sales would be half their original values.Diversification75Chapter 1Reducing RisknIf it were hot,their expected income would be$15,000 from air conditioners and$6,000
46、 from heaters,or$21,000.nIf it were cold,their expected income would be$6,000 from air conditioners and$15,000 from heaters,or$21,000.Diversification76Chapter 1Reducing RisknWith diversification,expected income is$21,000 with no risk.Diversification77Chapter 1Reducing RisknFirms can reduce risk by d
47、iversifying among a variety of activities that are not closely related.Diversification78Chapter 1Reducing RisknDiscussion QuestionslHow can diversification reduce the risk of investing in the stock market?lCan diversification eliminate the risk of investing in the stock market?The Stock Market79Chap
48、ter 1Reducing RisknRisk averse are willing to pay to avoid risk.nIf the cost of insurance equals the expected loss,risk averse people will buy enough insurance to recover fully from a potential financial loss.Insurance80Chapter 1The Decision to InsureNo$40,000$50,000$49,000$9,055Yes49,00049,00049,00
49、00InsuranceBurglary No Burglary Expected Standard(Pr=.1)(Pr=.9)Wealth Deviation81Chapter 1Reducing RisknWhile 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.nPurchases of insurance
50、 transfers wealth and increases expected utility.Insurance82Chapter 1Reducing RisknAlthough single events are random and largely unpredictable,the average outcome of many similar events can be predicted.The Law of Large Numbers83Chapter 1Reducing RisknExampleslA single coin toss vs.large number of c