Property 1: V(σ, μ) is concave if u(x) is concave.
Property 2: S
u
(σ, μ) ≥ S
v
(σ, μ) for all (σ, μ) if u(x) is more risk averse than v(x).
Property 3:
∂S
∂µ
≤
0
if u(x) is decreasingly absolute risk averse.
One reason why focusing on random variables from a location and scale family is a
worthwhile exercise is that for decision models with only one random variable, and whose
outcome variable is linear in that random variable, all possible random outcome variables
automatically form a location and scale family. For instance in the single risky asset portfo-
lio choice model where the outcome variable
W=W
0
(
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