1Chapter 17UncertaintyMain topics1. degree of risk2. decision making under uncertainty3. avoiding risk4. investing under uncertaintyDegree of risk • probability: number, 4, between 0 and 1 that indicates likelihood a particular outcome occur• frequency: estimate of probability, 4 = n/N, where n is number of times a particular outcome occurred during N number of times event occurred• if we don’t have frequency, may use subjective probability – informed guessProbability distribution• relates probability of occurrence to each possible outcome• first of two following examples is less certainProbability, %201040Days of rain per month0123410% 20% 10%30(a) Less Certain20% 40%Figure 17.1 Probability DistributionProbability, %201040Days of rain per month0123430% 40% 30%Probabilitydistribution30(b) More CertainExpected value example• 2 possible outcomes: rains, does not rain• probabilities are ½ for each outcome• promoter’s profit is • $15 with no rain• -$5 with rain• promoter’s expected value (“average”)EV = [Pr(no rain)GValue(no rain)]+[Pr(rain)GValue (rain)]= [ ½G $15] + [ ½ G (-$5)] = $52Variance and standard deviation• variance: measure of risk• variance = [Pr(no rain) G (Value(no rain - EV)2] + [Pr(rain) G (Value (rain) – EV)2]= [½G ($15 - $5)2] + [½ G (-$5 - $5)2]= [½G ($10)2] + [½ G (-$10)2] = $100• standard deviation = square root of varianceDecision making under uncertainty• a rational person might maximize expected utility: probability-weighted average of utility from each possible outcome• promoter’s expected utility from an indoor concert isEU = [Pr(no rain) G U(Value(no rain))] + [Pr(rain) G U(Value(rain))]= [½ G U($15)] + [½ G U(-$5)]• promoter’s utility increases with wealthFair bet• wager with an expected value of zero• flip a coin for a dollar:[½ G (1)] + [½ G (-1)] = 0Attitudes toward risk• someone who is risk averse is unwilling to make a fair bet • someone who is risk neutral is indifferent about making a fair bet• someone who is risk preferringwants to make a fair betRisk aversion• most people are risk averse: dislike risk• their utility function is concave to wealth axis: utility rises with wealth but at a diminishing rate• they choose the less risky choice if both choices have the same expected value• they choose a riskier option only if its expected value is sufficiently higher than a riskless one• risk premium: amount that a risk-averse person would pay to avoid taking a risk3Figure 17.2 Risk AversionUtility, UWealth, $10 26 40 64 70abdeU (W ealth)U ($70) = 1400.1U ($10) + 0.9U ($70) = 133U ($26) = 105U ($40) = 120U($10) = 700Risk premium0.5U ($10) + 0.5U($70) =cfRisk averse decision• Irma’s initial wealth is $40• her choice• she can do nothing: U($40) = 120• she may buy a risky Ming vaseExpected value of Ming Vase• worth $10 or $70 with equal probabilities• expected value (point d):$40 = [½ G $10] + [½ G $70]• expected utility (point b):105 = [½ G U($10)] + [½ G U($70)]Irma’s risk premium• amount Irma would pay to avoid this risk• certain utility from wealth of $26 is U($26) = 105• Irma is indifferent between • having the vase• having $26 with certainty• thus, Irma’s risk premium is $14 = $40 - $26 to avoid bearing risk from buying the vaseFigure 17.3a Risk NeutralityUtility, UWealth, $10 40 70abU (Wealth)(a) Risk-Neutral IndividualU ($70) = 140U ($10) = 700U ($40) = 1050.5U($70) =0.5U ($10) +cRisk-neutral person’s decision• risk-neutral person chooses option with highest expected value, because maximizing expected value maximizes utility• utility is linear in wealth105 = [½ G U($10)] + [½ G U($70)] = [½ G 70] + [½ G 140]• expected utility = utility with certain wealth of $40 (point b)4Figure 17.3b Risk PreferenceUtility, UWealth, $10 40 58 70abdecU (Wealth)(b) Risk-Preferring IndividualU ($70) = 140U ($40) = 82U ($10) = 7000.5U ($70) = 1050.5U ($10) +Risk-preferring person’s decision• utility rises with wealth• expected utility from buying vase, 105 at b, is higher than her certain utility if she does not vase, 82 at d• a risk-preferring person is willing to pay for the right to make a fair bet (negative risk premium)• Irma’s expected utility from buying vase is same as utility from a certain wealth of $58, so she’d pay $18 for right to “gamble”Risky jobs• some occupations have more hazards than do others• in 1995, deaths per 100,000 workers was • 5 across all industries• 20 for agriculture(35 in crop production)• 25 for miningRisk of workplace homicides per 100,000 workers1.3Fire fighter1.5Butcher-meatcutter2.3Bartender5.9Gas station worker6.1Police, detective10.7Sheriff-bailiff22.7Taxicab driverRateOccupationRisky jobs have small premium • Kip Viscusi found workers received a risk premium (extra annual earnings) for job hazards of $400 on average in 1969• amount was relatively low because annual risks incurred by workers were relatively small• in a moderately risky job,• danger of dying was about 1 in 10,000• risk of a nonfatal injury was about 1 in 100Value of life• given these probabilities, estimated average job-hazard premium implies that workers placed a value on their lives of about $1 million• and an implicit value on nonfatal injuries of $10,0005GamblingWhy would a risk-averse person gamble where the bet is unfair?• enjoys the game• makes a mistake: can’t calculate odds correctly• has Friedman-Savage utilityApplication GamblingUtility, UWealthW1W2W3W4W5abb* d *cdeU (Wealth)Avoiding risk• just say no: don’t participate in optional risky activities• obtain information•diversify • risk pooling• diversification can eliminate risk if two events are perfectly negatively correlatedPerfectly negatively correlated• 2 firms compete for government contract• each has an equal chance of winning• events are perfectly negatively correlated: one firm must win and the other must lose• winner will be worth $40• loser will be worth $10If buy 1 share of each for $40• value of stock shares after contract is awarded is $50 with certainty• totally diversified: no variance; no riskIf buy 2 shares of 1 firm for $40• after contract is awarded, they’re worth $80 or $20• expected value:$50 = (½ ´ $80) + (½ ´ $20) • variance:$900 = [½ ´ ($80 - $50)2] + [½ ´ ($20 - $50)2]• no diversification (same result if buy two stocks that are perfectly positively
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