Unformatted text preview: b
,
1 − bx b>0 rR (x ) = bx
1 − bx This utility function satisfies u ′ > 0 only when x < 1 / b , which implies that there
are certain states where the investor would choose less wealth over more. This is
a problem with the quadratic utility function RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 12 • Exponential utility function
1
u(x ) = − e −λx , λ ≠ 0
λ
rA (x ) = λ, rR (x ) = λx This utility function is referred to as constant absolute risk aversion (CARA)
because the absolute risk aversion is constant
• Power utility function
u(x ) = x α ,
rA (x ) = 0 < α <1 1−α
,
x rR (x ) = 1 − α This utility function is referred to as constant relative risk aversion (CRRA)
because the relative risk aversion is constant RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 13 • Logarithmic utility function u(x ) = ln(x )
rA (x ) = 1
,
x rR (x ) = 1 This utility function is part of the CRRA family. The logarithmic utility function
is an extension of the power utility function for that case α = 0 by the nature of xα −1
lim
= ln(x )
α→0
α RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.
 Fall '14

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