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Chapter 6 It is a truth very certain that when it is not in our power to determine what is true we ought to follow what is probable.— Descartes Decision Making and Risk: Certainty Equivalents 1 and Utility Decision Making Using Certainty Equivalents The certainty equivalent (CE) is the payoff amount we would accept in lieu of under- going the uncertain situation. Shirley Smart would pay $25 to insure her 1983 Toyota against total theft loss. CE = – $25. For $1,000, Willy B. Rich would sell his Far- Fetched Lottery rights. CE = $1,000. Lucky Chance would pay somebody $100 to fill 2 her Far-Fetched Lottery contract. CE = -$100. Risk Premiums A situation’s risk premium (RP) is the difference its between expected payoff (EP) and certainty equivalent (CE): RP = EP - CE Shirley Smart’s car is worth $1,000 and there is a 1% chance of its being stolen. Thus, going without insurance has EP = (– $1,000)(.01) + ($0)(.99) = – $10 RP = – $10 – (– $25) = $15 3 Risk Premiums Playing the Far-Fetched Lottery has EP = $2,500. Thus, For Willy B. Rich, RP = EP – CE = $2,5000 – ($1,000) = $1,500 For Lucky Chance, RP = EP – CE = $2,5000 – (– $100) = $2,600 Different people will have different CEs and RPs for the same circumstance. They have different attitudes toward risk. 4 Attitude Toward Risk People with positive RPs are risk averters. Lucky Chance has greater risk aversion than Willy B. Rich, as reflected by her greater RP. We cannot compare Shirley’s risk aversion to the others’ because circumstances differ. Risk averse persons have RPs that increase: When the downside amounts become greater. Or when the chance of downside increases. A risk seeker will have negative RP. 5 A risk neutral person has zero RP. Maximizing Certainty Equivalent A plausible axiom: Decision makers will prefer the act yielding greatest certainty equivalent. A logical conclusion: The ideal decision criterion is to maximize certainty equivalent. Doing so guarantees taking the preferred action. But CEs are difficult to determine. One approach is to discount the EPs. RP = EP – CE implies that CE = EP – RP. 6 Using Risk Premiums to Get Certainty Equivalents Ponderosa Records president has the following risk premiums, keyed to the downside. These were found by extrapolating from three equivalencies (white boxes). Exact amounts are unknowable, but these values seem 7 to fit his risk profile. Decision Tree Analysis with CEs (Discounted Expected Payoffs) 8 How Good is the Analysis? This result is different from that of ordinary back folding (Bayes decision rule). It specifically reflects underlying risk aversion. The result must be correct if CEs are right. The major weakness is the ad hoc manner for getting the RPs, and hence the CEs. Many assumptions are made in extrapolating to get the table of RPs. There is a cleaner way to achieve the same 9 thing using utilities. Utility Theory Consider a set of outcomes, O1, O2, ..., On. The following assumptions are made: Preference ranking can be done. Transitivity of preference: A is preferred to B and B to C, then A must be preferred to C. Continuity: Consider Obetween. Take a gamble between two more extreme outcomes; winning yields Obest and losing Oworst. There is a win probability q making you indifferent between getting Obetween and gambling. Such a gamble is 10 called a reference lottery. Utility Assumptions Continuity (continued): e.g., +$1,000 v. Far-Fetched Lottery, you pick q. For Willy B. Rich, q = .5. (His CE was = +$1,000.) For Lucky Chance, q = .9. If the win probability were .99, would you risk +$1,000 to gamble? What is your q? Substitutability: In a decision structure, you would willingly substitute for any outcome a gamble equally preferred. One outcome on Lucky Chance’s tree is +$1,000; she would accept substituting for it the Far-Fetched 11 Lottery gamble with .9 win probability. Utility Assumptions and Values Increasing preference: Raising q makes any reference lottery more preferred. Anybody would prefer the revised Far-Fetched Lottery when two coins are tossed and just one head will win the $10,000. (The win probability goes from .5 to .75.) You still might not like that gamble! Outcomes can be assigned utility values arbitrarily, so that the more preferred always gets the greater value: u(Obest) = 10 u(Oworst)=0 u(Obetween)=50 Willy has u(+$10,000) = 500, u(-$5,000) = 0 and 12 u(+$1,000) = 250. These are his values only. Utility Values Lucky has different values: u(+$10,000) = 50, u(-$5,000) = -99, and u(+$1,000) = 35.1. Like temperature, where 0o and 100o are different states on Celsius and Fahrenheit scales, so utility scales may differ. The freezing point for water is 0o C and the boiling point 100o C. In-between states will have values in that range, and hotter days will have greater temperature values than cooler. So, too, with utility values. They will fall into the range defined by the extreme outcomes, Oworst and Obest. More preferred outcomes will 13 have greater utilities Utility Values A reference lottery can be used to find the utility for an outcome Obetween by: First, establish an indifference win probability qbetween making it equally preferred to the gamble: Obest with probability qbetween and Oworst with probability 1 - qbetween Second, compute the lottery’s expected utility: u(Obest)(qbetween) + u(Oworst)(1 - qbetween) The above is u(Obetween). 14 Utility Values Using the Far-Fetched Lottery as reference: Lottery Willy Lucky Outcomes Prob. Utility Prob. Utility Obest (+$10,000) q=.5 500 q=.9 50 Oworst (-$5,000) 1 -.5 0 1 -.9 -99 Expected Utility: 250 35.1 Obetween (+$1,000): 250 35.1 The indifference q plays a role analogous to the thermometer, reading attitude towards the 15 outcome similarly to measuring temperature. The Utility Function Utility values assigned to monetary outcomes constitute a utility function. From a few points we may graph the utility function and apply it over a monetary range. Those points may be obtained from an interview posing hypothetical gambles. Using u(+$10,000)=100 and u(-$5,000)=0 Shirley Smart gave the following equivalencies: A: +$10,000 @ qA v -$5,000 ≡ +$1,000 if qA =.70 B: +$10,000 @ qB v +$1,000 ≡ +$5,000 if qB =.75 C1: +$1,000 @ qC1 v -$5,000 ≡ -$500 if qC1 =.70 16 C2: +$1,000 @ qC2 v -$5,000 ≡ -$2,000 if qC2 =.30 Shirley’s Utility Function Shirley’s utilities for the equivalent amounts are equal to the respective expected utilities: u(+$1,000) = u(+$10,000)(.70) + u(-$1,000)(1-.70) = 100(.70) + 0(1-.70) = 70 u(+$5,000) = u(+$10,000)(.75) + u(+$1,000)(1-.75) = 100(.75) + 70(1 - .75) = 92.5 u(-$500) = u(+$1,000)(.70) + u(-$5,000)(1-.70) = 70(.70) + 0(1 - .70) = 49 u(-$2,000) = u(+$1,000)(.30) + u(-$5,000)(1-.30) = 70(.30) + 0(1 - .30) = 21 Altogether, Shirley gave 6 points, plotted on the following graph. The smoothed curve fitting 17 through them defines her utility function. Shirley’s Utility Function 18 Using the Utility Function This utility function applies to the Ponderosa decision. 19 Using the Utility Function Read the utility payoffs corresponding to the net monetary payoffs. Apply the Bayes decision rule, with either: A utility payoff table, computing the expected payoff each act. Or a decision tree, folding it back. The certainty equivalent amount for any act or node may be found from the expected utility by reading the curve in reverse. The following Ponderosa Records decision 20 tree was folded back using utility payoffs. Decision Tree Analysis with Utilities 21 Shape of Utility Curve and Attitude Toward Risk The following shapes generally apply. The risk averter has decreasing marginal util- ility for money. He will buy casualty insurance and losses weigh more heavily than like gains. Risk seekers like some unfavorable gambles. 22 Risk neutrality values money at its face amount. Important Utility Ramifications Hybrid shapes (like Shirley’s) imply shifting attitudes as monetary ranges change. Regardless of shape, maximizing expected utility also maximizes certainty equivalent. Therefore, applying Bayes decision rule with utility payoffs discloses the preferred action. Primary impediments to implementation: Clumsiness of the interview process. Multiple decision makers. Attitudes change with circumstances and time. 23 Ratification of Bayes Decision Rule Over narrow monetary ranges, utility curves resemble straight lines. For a straight line, expected utility equals the utility of the expected monetary payoff. Maximizing expected monetary payoff then also maximizes expected utility. Thus: The Bayes decision rule discloses the preferred action as long as the outcomes are not extreme. Managers can then delegate decision making without having to find utilities. 24 Preferred actions will be found by the staff. Using Utility Functions with PrecisionTree PrecisionTree can be used to evaluate decision trees with with exponential and logarithmic utility functions. To get started, click on the name box of a decision tree and the Tree Setting dialog box appears, as shown next. A B 1 tree #1 1 2 0 25 Tree Settings Dialog Box 2. Select the type of utility (Figure 6-14) function in the Function line. 1. Check the Here exponential Use Utility is chosen. Function box. 3. Select the risk coefficient, R, in the R value line. Here 10,000 is used. 4. Select Expected Utility in the Display line. Other options are Certainty Equivalent and Expected Value. 5. Click OK. 26 Decision Tree with Exponential Utility A Function for R = 10,000 (Figure 6-15) B C D E F 1 Success 80.0% 0.0 The2 optimal strategy is: $90,000 1 3 Market nationally FALSE 4 1. Not test market and to -$50,000 -48 5 Failure 20.0% 0.0 abort. 6 $0 -244 7 Favorable 50.0% 2. The corresponding 8 $10,000 -1 9 TRUE 0 expected utility is 0. 10 Abort $0 -1 11 Test market FALSE 12 -$15,000 -2 13 Success 20.0% 0.0 14 $90,000 1 15 Market nationally FALSE 16 -$50,000 -531 17 Failure 80.0% 0.0 18 $0 -664 19 Unfavorable 50.0% 20 $0 -3 21 Abort TRUE 0 22 $0 -3 23 Ponderosa Record Company 24 0 25 Success 50.0% 0 26 $100,000 1 27 Market nationally FALSE 28 -$60,000 -201 29 Failure 50.0% 0 30 $0 -402 31 Don't test market TRUE 32 $0 0 33 27 Abort TRUE 1 34 $0 0

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posted: | 4/21/2011 |

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