eco331-05-RiskAversionHandout

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

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Unformatted text preview: Overview Risk Aversion Comparing Risk Attitudes Terminal Wealth Eco331: Risk Aversion Marciano Siniscalchi October 12, 2009 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 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 = all discrete r.v.s with values in X . We also assume F F . Preference < on F or F . < 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. Overview Risk Aversion Comparing Risk Attitudes Terminal Wealth Certainty Equivalent Definition For any x X , x F is the r.v. with distribution ( x , 1). Definition For any r.v. X F , x X is a certainty equivalent of X if 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 admits a unique certainty equivalent , denoted C [ X ] . 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 is risk-averse iff, for any X F , E [ X ] < X . Risk-loving : E [ X ] 4 X . Risk-neutral : E [ X ] X . Note : an individual may be neither risk-loving nor risk-averse. 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 E [ X ] < 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 , C [ X ] E [ X ] ....
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This note was uploaded on 11/17/2009 for the course ECONOMICS 331 taught by Professor Marciano during the Spring '09 term at Northwestern.

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

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