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 will 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• rational person maximizes 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 unwilling to make a fair bet is risk averse• someone who is indifferent about making a fair bet is risk neutral• someone who wants to make a fair bet is risk preferringRisk aversion• most people are risk averse: dislike risk• they will choose a riskier bundle only if its expected value is much higher• their utility function is concave to wealth axis: utility rises with wealth but at a diminishing rate• risk premium: amount that a risk-averse person would pay to avoid taking a riskFigure 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) =cf3Figure 17.3a Risk NeutralityUtility, UWealth, $10 40 70abU (Wealth)(a) Risk-Neutral IndividualU ($70) = 140U ($10) = 700U ($40) = 1050.5U($70) =0.5U ($10) +cFigure 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) +GamblingWhy 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 $104If 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 correlated)If stocks values are uncorrelated• each firm has 50% chance of a government contract• whether a firm gets a contract doesn’t affect whether other wins one• expected value$50 = (¼ ´ $80) + (½ ´ $50) + (¼ ´ $20) • variance$450 = [¼ ´ ($80 - $50)2] + [½ ´ ($50 - $50)2] + [¼´ ($20 - $50)2]• buying both results in some diversificationMutual funds• provide some diversification• Standard & Poor’s Composite Index of 500 Stocks (S&P 500)• Wilshire 5000 Index Portfolio (actually 7,200 stocks)Insurance• risk averse people will pay money – risk premium – to avoid risk• world-wide insurance premiums in 1998: $2.2 trillionHouse insurance• Scott is risk averse• wants to insure his $80 (thousand) house• 25% chance of fire next year• if fire occurs, house worth $405With no insurance• expected value of house is$70 = (¼ ´ $40) + (¾ ´ $80) • variance$300 = [¼ ´ ($40 - $70)2] + [¾ ´ ($80 - $70)2]With insurance• suppose insurance company offers fair insurance• lets Scott trade $1 if no fire for $3 if fire• insurance is fair bet because expected value is$0 = (¼ ´ [-$3]) + (¾ ´ $1)• Scott fully insurances: eliminates all risk• pays $10 if no fire• receives $30 if fire• net wealth in both states of nature is $70Commercial insurance • is not fair• available only for diversifiable risksInvesting under uncertainty• monopoly’s owner has an uncertain payoff this year• if risk neutral, owner maximizes expected value of return• otherwise, owner maximizes his or her expected utility• summarize analysis in decision treeFigure 17.04 Investment Decision Tree with Risk AversionLow demandHigh demand$20080%20%–$100$0EV=$140EV=$140Invest(a) Risk-Neutral OwnerDo not investLow demandHigh demandU($200) = 4080%20%U(–$100) = 0U($0) = 35EU =35EU=32Invest(b) Risk-Averse OwnerDo not investInvesting under uncertainty and discounting• problem is more complicated if future returns are uncertain• need to calculate expected utility (or value) and then discount6Figure 17.5 Investment Decision Tree with Uncertainty andDiscountingLow demandHigh demandR = $125C = $2580%20%R = $50$0ENVP=$75EV=$110EPV = $100InvestThis yearNext yearDo not investInvesting with advertising• future demand is uncertain• advertising affects demand• suppose risk neutral ownerFigure 17.6 Investment Decision Tree with AdvertisingEV=$10InvestDo
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