Presentation on theme: "Risk Attitude Dr. Yan Liu"— Presentation transcript:
1Risk Attitude Dr. Yan Liu Department of Biomedical, Industrial & Human Factors EngineeringWright State University
2Which game would you choose, game 1 or game 2? IntroductionGame 1(0.5)Payoff$30-$1$2,000-$1,900Game 2EMV=$14.5Which game would you choose, game 1 or game 2?EMV=$50If EMV is the basis for the decision, you should choose Game 2. Most of us, however, may consider Game 2 to be too risky and thus choose Game 1.This example illustrates that EMV analysis does not capture risk attitudes of decision makers. Individuals who are afraid of risk or are sensitive to risk are called risk-averse.
3Utility FunctionUtility functions are models of an individual’s attitude toward riskUtility functions translate dollars into utility unitsGraphTableMathematical expressionUtilityDollarsxU(x)A utility function that displays risk-aversion(upward sloping and concave)
4Risk Attitude Risk-Averse: Afraid or Sensitive to Risk Would trade a gamble for a sure amount that is less than the expected value of the gambleU(x) is a concave curve(continuous)(discrete)Risk-Seeking: Willing to Accept More RiskWould play a state lotteryU(x) is a convex curve(continuous)(discrete)Risk-Neutral: An EMV Decision MakerMaximizing utility is the same as maximizing EMVU(x) is a straight lineis constant (continuous)(discrete)
5Risk Attitude (Cont.)DollarsUtilityRisk-AverseRisk-NeutralRisk-SeekingShapes of Utility Functions of Three Different Risk Attitudes
6Some Terminologies Expected Utility (EU) Certainty equivalent (CE) Weighted average of utilities of all possible statesCertainty equivalent (CE)Amount of money equivalent to the situation that involves uncertaintyRisk PremiumDifference between the EMV and the CE, i.e., the amount you would pay to avoid the riskYou have a lottery which has 0.3 probability of winning $200 and 0.7 probability of losing $10, and you are willing to sell it for $30.Your certainty equivalent for this lottery is $30The risk premium of the lottery is:(0.3* *(-10))-30 =53-30=$23
7Certainty Equivalent (CE) EMVRisk PremiumUtility CurveExpected Utility (EU)UtilityDollarU(CE) = EUGraphical Representation of Expected Utility, Certainty Equivalent, and Risk PremiumFor a risk-seeking person, CE would be on the right side of EMV on the horizontal axis
8Utility AssessmentAssessing a Utility Function is a Subjective JudgmentDifferent people have different risk attitudes toward risk and are willing to accept different levels of riskTwo MethodsAssessment using certainty equivalentAssessment using probabilities
9Utility Function Assessment Via Certainty Equivalence Assess several certainty equivalents from which the utility function is derivedStep 1: set the utility of the best payoff to 1 and the utility of the worst payoff to 0Step 2: Construct a situation that involves uncertainty and find its CE using reference lottery.Step 3: Calculate the expected utility of the lottery, EU. Because EU is equal to U(CE), we get another point (CE, EU) on the utility curveStep 3: Repeat Steps 2 and 3 until getting enough points to plot the utility curve
10Suppose your CE in this lottery is $30 You face an uncertain situation in which you may earn $10 in the worst case, $100 in the best case, or some amount in between. You have a variety of options, each of which leads to some uncertain payoff between $10 and $100. To evaluate the alternatives, you must assess your utility for payoffs from $10 to $100.Step 1: let U(10)=0 and U(100)=1Step 2: imagine you have the opportunity to play the following reference lotteryA(0.5)$100$10BCESuppose your CE in this lottery is $30Step 3: Calculate EU of lottery A, which is 0.5∙U(100)+0.5∙U(10)=0.5. Therefore, U(30)=0.5 and you have found a third point on your utility curve.Step 4: To find another point, you can take a different reference lottery, say using $100 and $30 as two equally likely outcomes in lottery A, and then follow steps 2 and 3. Continue with the same procedures until you have enough points to plot the utility function.
11Suppose you now have five points on your utility curves: U(10)=0, U(18)=0.25, U(30)=0.5, U(50)=0.75, and U(100)=1, you can plot the utility functionx (Dollars)U(x)
12Utility Function Assessment Via Probability-Equivalent Assess the utility of a selected dollar amount directlyAdjust the probability in the reference lotteryC(p)(1-p)$100$10D$65U(65) = p∙U(100) + (1-p)∙U(10) = p∙1+(1-p)∙0 = p
13Gambles, Lotteries, and Investments Framing utility assessment in terms of gambles or lotteries may evoke images of carnival games or gambling which seem irrelevant to decision making or even distastefulAn alternative is to think in terms of risky investment, particularly for investment decisionsWhether you should make a particular investment
14Risk Tolerance and Exponential Utility Function R is risk tolerance, showing how risk-adverse the function is. Larger R means less risk-adversion and makes the utility function flatterExponential Utility Functions with Three Different Risk Tolerancesx ↑ => U(x) →1x =0 => U(x) = 0
15Risk Tolerance and Exponential Utility Function (Cont.) Assess Risk Tolerance RE(0.5)$Y– $Y/2F$0The largest Y for which you prefer to take gamble E is approximately equal to your risk toleranceSuppose you decide Y is $900,then R=900, and your utility function is
16Risk Tolerance and Exponential Utility Function (Cont.) Find CE of Given Uncertain EventFirst calculate the expected utility (EU) of the uncertain eventSince U(CE)=EU, you can solve the equation to get CEIf you estimate the expected value, ,and variance, , of the payoffs, then CE can be approximately calculated as
17Suppose you face the following gamble: 1) win $2000 with probability 0 Suppose you face the following gamble: 1) win $2000 with probability 0.4; 2) win $1000 with probability 0.4, or 3) win $500 with probability 0.2, and your utility can be modeled as an exponential function with R=900. What is your CE of this gamble?The expected utility of the gamble is:EU = 0.4∙U($2000)+0.4 ∙U($1000)+ 0.2∙U($500)= 0.4∙(1-e-2000/900) +0.4 ∙(1-e-1000/900)+ 0.2∙(1-e1-500/900) = 0.710Solve 0.710=1-e-CE/900 for CE, you can get CE=$
18Constant and Decreasing Risk-Aversion Constant Risk-AversionThe risk premium for a gamble does not depend on the initial wealth held,Can be represented using an exponential utility functionYou have $x in your pocket, and you are facing a bet: 1) win $15 with probability 0.5, or 2) lose $15 with probability 0.5. Suppose your utility function can be modeled as an exponential function with risk tolerance R=35.(0.5)$x+15$x –15$ in PocketProbability Tree of the Bet
20Constant and Decreasing Risk-Aversion (Cont.) Typically, people’s attitude towards risk changes with their initial wealthThe risk premium for a gamble decreases along with the increase of wealth heldCan be represented with the logarithmic utility function
21In the previous example of betting, suppose your utility function can be modeled as a logarithmic functionWhen x=$25EU=0.5∙U($10)+0.5∙U($40)=0.5∙ln(10)+0.5∙ln(40)=2.9957EU=U(CE) 2.9957=ln(CE)CE=$20EMV= 0.5∙10+0.5∙40=$25Risk Premium=EMV-CE=25-20=$5$xEV ($)CE ($)Risk Premium ($)25205.003531.623.384542.432.575552.922.08
22Some Caveats Utilities DO NOT Add Up U(A+B)≠U(A)+U(B) (why?)Utility Difference Does Not Express Strength of PreferencesU(A1)-U(A2) > U(B1)-U(B2) does not mean we would rather go from A1 to A2 instead of from B1 to B2Utility only provides a numerical scale for ordering preferences, not a measure of their strengthsUtilities are Not Comparable from Person to PersonA utility function is a subjective personal statement of an individual’s preference
23ExerciseAn investor with assets of $10,000 has an opportunity to invest $5,000 in a venture that is equally likely to pay either $15,000 or nothing. The investor’s utility function can be described by the log utility function U(x) =ln(x), where x is the total wealth.What should the investor do?
24a. Total Wealth success (0.5) 10,000-5,000+15,000 =$20,000 $15,000 Invest$10,000success (0.5)10,000-5,000= $5,000Don’t Invest10,000-5,000+15,000 =$20,000Failure (0.5)Total Wealth- $5,000$15,000EU(invest) = 0.5∙U($20,000)+0.5∙U($5,000)=0.5∙ln($20,000)+0.5∙ln($5000) = 9.21EU(Don’t invest) = U($10,000) = ln($10,000) = 9.21Therefore, the investor is indifferent between the two alternatives
25b. Suppose the investor places a bet with a friend before making the investment decision. The bet is for $1,000; if a fair coin lands heads up, the investor wins $1,000, but if it lands tails up, the investor pays $1,000 to his friend. Only after the bet has been resolved will the investor decide whether or not to invest in the venture. If he wins the bet, should he invest? What if he loses the bet? Should he toss the coin in the first place?
27If he wins the bet:EU(Invest) = 0.5∙ln($21,000) + 0.5∙ln($6,000) = 9.326EU(Don’t Invest) = ln($11,000) = 9.306Therefore, if he wins the bet, he should invest the ventureIf he loses the bet:EU(Invest) = 0.5∙ln($19,000) + 0.5∙ln($4,000) = 9.073EU(Don’t Invest) = ln($9,000) = 9.105Therefore, if he losses the bet, he should not invest the ventureEU(Bet) = 0.5∙EU(Invest|win) + 0.5∙EU(Don’t Invest |lose)= 0.5(9.326)+0.5(9.105) = 9.216EU(Don’t Bet) = 9.21 (from part a)Therefore, he should bet