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Lecture19-2010 - Eliciting Risk Aversion Coecients Lecture...

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Eliciting Risk Aversion Coefficients: Lecture XIX Charles B. Moss October 7, 2010 I. Eliciting Risk Aversion Coefficients A. Direct Elicitation of Utility Functions 1. Assume two possible outcomes of a random variable Y 1 which occurs with probability p and Y 2 which occurs with probability 1 P . 2. As the decision maker whether he prefers the risky alternative to a certain payoff ˜ Y . a) Suppose the decision maker is given an alternative be- tween a 50/50 bet of paying $50 or $5 (with the expected return of $27.50) or a certain payoff of $25. b) If the decision maker choose the risky alternative, the de- cision maker’s risk aversion coefficient is less than 0.005944 E [ U ( Y )] = exp ( ρY 1 ) p exp ( ρY 2 ) (1 p ) = exp ρ ˜ Y (1) c) The value for ρ can be solved using Gauss-Seidel exp ( ρ 26 . 0) 0 . 50 exp ( ρ 5) 0 . 50 exp ( ρ 50) = 0 . (2) 2. If the answer is no (they prefer the certain outcome to the risky alternative), either 1
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AEB 6182 Agricultural Risk Analysis and Decision Making Professor Charles B. Moss Lecture XIX Fall 2010 a) Change the probabilities by making the higher return more likely (i.e., let p = 0 . 40), exp ( ρ 26 . 0) 0 . 40 exp ( ρ 5) 0 . 60 exp ( ρ 50) = 0 (3) for which the risk aversion coefficient increases to 0.0113. b) Alternatively, decrease the certainty equivalent (i.e., to $25.75). B. Equally Likely Risky Prospect and Finding its Certainty Equiva- lent (ELCE) 1. This process begins by assigning the expected utility at two endpoint outcomes. a) Consider a low income of $50,000 ( a = 50 , 000) and a high income of $100,000 ( b = 100 , 000). b) Next, assign a utility value at each endpoint ( U (50 , 000) = U ( a ) = 0 and U (100 , 000) = U ( b ) = 1). 2. The research then asks the decision maker how much they would be willing to give or take (or what is the certainty equivalent) for a gamble paying $50,000 with probability 0.50 and $100,000 with probability 0.5.
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