Digital Signal Processing Reference
In-Depth Information
F
F
u
(ω
)
≈
u
(
E
W
∼
p
ψ
[
ω
]
)
x
=
E
W
∼
p
ψ
[
ω
]
)
×
∂
∂
F
F
+
(ω
−
E
W
∼
p
ψ
[
ω
x
u
(
x
)
F
]
x
=
E
W
∼
p
ψ
[
ω
F
F
2
+
(ω
−
E
W
∼
p
ψ
[
ω
]
)
2
×
∂
x
2
u
(
x
)
,
2
∂
F
]
(15.24)
F
F
u
(
E
W
∼
p
ψ
[
ω
]−
p
ψ
(
A
;
Θ
))
≈
u
(
E
W
∼
p
ψ
[
ω
]
)
x
=
E
W
∼
p
ψ
[
ω
;
Θ
)
×
∂
∂
−
p
ψ
(
A
x
u
(
x
)
.
(15.25)
F
]
If we take expectations of (
15.24
) and (
15.25
), simplify taking into account the
fact that E
W
∼
p
ψ
[
ω
F
F
0, and match the resulting expressions using
(
15.23
), we obtain the following approximate expression for the risk premium:
−
E
W
∼
p
ψ
[
ω
]]=
1
2
Va r
W
∼
p
ψ
[
ω
F
p
ψ
(
A
;
Θ
)
≈
]
⎧
⎨
⎩
−
∂
−
1
⎫
x
=
E
W
∼
p
ψ
[
ω
x
=
E
W
∼
p
ψ
[
ω
⎬
2
∂
∂
×
x
2
u
(
x
)
x
u
(
x
)
⎭
∂
F
F
]
]
r
(
p
ψ
)
.
(15.26)
Thus, approximation “in the small” of the risk premium makes it proportional to
the variance of rewards
and
function
r
(
p
ψ
)
, which is just, in the language of risk
aversion, the Arrow-Pratt measure of
absolute risk aversion
[
9
,
23
]. This expression
for the risk premium is obviously not the one we shall use: its purpose is to shed light
on the measure of absolute risk aversion, and derive the expression of
u
,asshown
in the following Lemma.
Lemma 3.
r
(
p
ψ
)
=
k, a constant matrix iff one of the following conditions holds
true:
u
(
x
)
=
x
if
k
=
0
.
(15.27)
u
(
x
)
=−
exp
(
−
ax
)
for
some a
∈ R
∗
(otherwise)
The proof of this Lemma is similar to the ones found in the literature (
e.g.
[
9
], Chap. 4).
The framework of Lemma 3 is that of
constant absolute risk aversion
(CARA) [
9
],
the framework on which we focus now, assuming that the investor is risk-averse.
This implies
k
0; this constant
a
is called the
risk-aversion parameter
,
and shall be implicit in some of our notations. We obtain the following expressions
for
=
0 and
a
>
c
ψ
and
p
ψ
.
