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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 = decisionmaker. 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 riskaverse iff, for any X F , E [ X ] < X . Riskloving : E [ X ] 4 X . Riskneutral : E [ X ] X . Note : an individual may be neither riskloving nor riskaverse. Overview Risk Aversion Comparing Risk Attitudes Terminal Wealth Two Easy Consequences of the Definition Fact An EU DM with strictly increasing Bernoulli utility is riskneutral if and only if she is an expectedvalue maximizer . Proof: if riskneutral, 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.
 Spring '09
 Marciano
 Economics

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