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4. MeanVariance Analysis and Portfolio
Separation
4.1. Motivation
Accepting the expected utility maximization framework, we now discuss the conditions for
meanvariance analysis to be valid. An individual’s utility function may be expanded as a
Taylor series around his expected endofperiod wealth (assume the Taylor series converges):
u
(
f
W
)=
u
(
E
(
f
W
)) +
u
0
(
E
(
f
W
))(
f
W
−
E
(
f
W
)) +
1
2
u
00
(
E
(
f
W
))(
f
W
−
E
(
f
W
))
2
+
R
3
where
R
3
=
P
∞
n
=3
1
n
!
u
(
n
)
(
E
(
f
W
))(
f
W
−
E
(
f
W
))
n
and
u
(
n
)
(
·
)
is the
n
th derivative of
u
(
·
)
.S
incethe
expectation and summation operators are interchangeable, the individual’s expected utility
can be expressed as
E
(
u
(
f
W
)) =
u
(
E
(
f
W
)) +
1
2
u
00
(
E
(
f
W
))
σ
2
(
f
W
)+
E
(
R
3
)
(4.1)
where
E
(
R
3
∞
X
n
=3
1
n
!
u
(
n
)
(
E
(
f
W
))
m
n
(
f
W
)
(4.2)
and
m
n
(
f
W
E
((
f
W
−
E
(
f
W
))
n
)
is the
n
th central moment of
f
W
. Note that equation (4.1)
indicates a preference for expected wealth and an aversion to variance of wealth for an individ
ual having an increasing and strictly concave utility function. However, equation (4.2) shows
that expected utility cannot be de
f
ned solely over the expected value and variance of wealth
for arbitrary distributions and preferences, as indicated by the remainder term which involves
higher order terms. Therefore, to motivate the meanvariance model, we need at least one of
the following:
1. Quadratic preferences (for arbitrary distributions):
u
(
f
W
f
W
−
b
2
f
W
2
Under quadratic preferences, the third and higher order derivatives of
u
(
·
)
are zero.
That is,
E
(
R
3
)=0
and
E
(
u
(
f
W
)) =
E
(
f
W
)
−
b
2
E
(
f
W
2
E
(
f
W
)
−
b
2
³
(
E
(
f
W
))
2
+
σ
2
(
f
W
)
´
Therefore, when
E
(
f
W
)
and
σ
2
(
f
W
)
are
f
nite, quadratic utility is su
ﬃ
cient (in fact,
necessary for arbitrary distributions) for meanvariance analysis.
1
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View Full Document Remark 1.
Some undesirable properties are:
a. Satiation: An increase in wealth beyond the satiation point decreases utility.
b. Increasing absolute risk aversion in wealth:
R
A
(
f
W
)
≡
−
u
00
(
f
W
)
u
0
(
f
W
)
=
b
1
−
b
f
W
dR
A
(
f
W
)
d
f
W
=
b
2
(1
−
b
f
W
)
2
>
0
This implies that risky assets are inferior goods.
2. Rates of return follow a
multivariate normal distribution
(for arbitrary preferences):
In general, a su
ﬃ
cient condition for meanvariance framework is that
(
µ, σ
)
can
com
pletely characterize
the feasible choices, and such twoparameter distributions are closed
(
stable
) under linear combinations, i.e.,
X,Y
∈
F
⇒
aX
+
bY
∈
F
Remark 2.
A disadvantage of using normal distribution is that it is unbounded from
below. This is inconsistent with limited liability and with economic theory, which at
tributes no meaning to negative consumption.
De
f
ne the relationship between wealth and return as
e
r
≡
f
W
−
W
0
W
0
=
f
W
W
0
−
1
with
µ
≡
E
(
e
r
)=
E
(
i
W
W
0
−
1)
and
σ
2
≡
σ
2
(
e
r
σ
2
(
i
W
)
W
2
0
. To show the representation of going
from
u
(
f
W
)
to
v
(
µ, σ
2
)
, we can write the utility function as
u
(
·
u
(
e
r
;
µ, σ
2
)
⇒
E
(
u
(
·
)) =
Z
∞
−∞
u
(
e
r
;
µ, σ
2
)
f
(
e
r
;
µ, σ
2
)
d
e
r
Standardize
e
r
by de
f
ning
e
z
=
h
r
−
µ
σ
.T
h
e
n
e
r
=
µ
+
σ
e
z
, which implies that
e
z
=
±
∞
when
e
r
=
±
∞
,and
d
e
r
=
σd
e
z
. By change of variable,
f
(
e
r
;
µ, σ
2
1
σ
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This note was uploaded on 01/15/2010 for the course FIN FIN4160 taught by Professor Prof.chow during the Fall '09 term at CUHK.
 Fall '09
 Prof.Chow
 The Land

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