Lecture06-2004 - Utility Functions, Risk Aversion...

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1 Utility Functions, Risk Aversion Coefficients and Transformations Lecture VI I. An examination of the Arrow-Pratt Coefficients for particular functions. A. Quadratic Utility Function: To specify the appropriate shape of the utility function, the quadratic function becomes ( ) () 2 2 2 U w aw bw Uw a b w Uw b =− ′′ Arrow-Pratt absolute risk aversion coefficient: 22 21 20 2 AA A bb Rw ab w w dR w f x bd b dw dx f x w f x = −−  => =   Arrow-Pratt relative risk aversion coefficient 2 2 2 2 2 R R w a w b w b dw a bw w ==    = +> B. Power Utility Function: 1 1 1 r r r w r Uw w r w − − = = Arrow-Pratt absolute risk aversion coefficient: 1 2 0 r A r A rw r ww r dw w = −= < Arrow-Pratt relative risk aversion coefficient: 1 0 r R r R rw w R wr w dw = = Constant relative risk aversion.
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2 C. Negative Exponential Utility Function: ( ) ( ) () ( ) 2 exp exp exp Uw w w w =− −ρ ′′ = ρ− ρ Arrow-Pratt absolute risk aversion coefficient () ( ) 2 exp exp 0 A A w Rw w dR w dw  −ρ −ρ = −= ρ  ρ  = Constant absolute risk aversion. Arrow-Pratt relative risk aversion coefficient ( ) 2 exp exp 0 R R w R ww w w dw −ρ −ρ = ρ ρ =ρ> D. HARA–Hyperbolic Absolute Risk Aversion: 1 1 2 2 2 1 ,0 1 1 11 1 1 1 aw b b aw a b aw ab aw a Uw a b aw γ
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Lecture06-2004 - Utility Functions, Risk Aversion...

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