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eco331-05-RiskAversionHandout

eco331-05-RiskAversionHandout - Overview Risk Aversion...

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Overview Risk Aversion Comparing Risk Attitudes Terminal Wealth Eco331: Risk Aversion Marciano Siniscalchi October 12, 2009
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Overview Risk Aversion Comparing Risk Attitudes Terminal Wealth Overview Review of definitions Risk Aversion: Behavioral Definition Risk Aversion and Concavity of Bernoulli Utility Comparing Risk Attitudes Terminal Wealth vs. Gains and Losses
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Overview Risk Aversion Comparing Risk Attitudes Terminal Wealth Review of Definitions Set X of prizes; today: X an “interval” in R Set F of random variables (as functions or distributions). Henceforth abbreviated r.v. (traditional). New notation (traditional): F 0 = all discrete r.v.’s with values in X . We also assume F 0 ⊂ F . Preference < on F or F 0 . < consistent with EU : X < Y iff E [ u ( X )] E [ u ( Y )]. Unless otherwise noted, we assume < is consistent with EU! More traditional terminology: DM = “decision-maker.”
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Overview Risk Aversion Comparing Risk Attitudes Terminal Wealth Certainty Equivalent Definition For any x ∈ X , δ x ∈ F 0 is the r.v. with distribution ( x , 1). Definition For any r.v. X ∈ F , x ∈ X is a certainty equivalent of X if δ x X . Proposition If < is an EU preference with a continuous and strictly increasing utility function u : X → R , and X is an interval in R , then every X ∈ F 0 admits a unique certainty equivalent , denoted C [ X ] .
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Overview Risk Aversion Comparing Risk Attitudes Terminal Wealth Risk Aversion: Formal Definition In the St. Petersburg Paradox, E [ X ] = , so ln( E [ X ]) = > E [ln( X )] : EU DM with u ( x ) = ln( x ) would rather get E [ X ] for sure than X . Definition A DM with preferences < on F 0 is risk-averse iff, for any X ∈ F 0 , δ E [ X ] < X . Risk-loving : δ E [ X ] 4 X . Risk-neutral : δ E [ X ] X . Note : an individual may be neither risk-loving nor risk-averse.
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Overview Risk Aversion Comparing Risk Attitudes Terminal Wealth Two Easy Consequences of the Definition Fact An EU DM with strictly increasing Bernoulli utility is risk-neutral if and only if she is an expected-value maximizer . Proof: if risk-neutral, X < Y iff δ E [ X ] < δ E [ Y ] iff E [ X ] E [ Y ] Fact An EU preference with continuous and strictly increasing utility u denotes risk aversion if and only if, for every X ∈ F 0 , C [ X ] E [ X ] .
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