In financial mathematics, a distortion risk measure is a type of risk measure which is related to the cumulative distribution function of the return of a financial portfolio.
The function
ρ
g
:
L
p
→
R
associated with the distortion function
g
:
[
0
,
1
]
→
[
0
,
1
]
is a distortion risk measure if for any random variable of gains
X
∈
L
p
(where
L
p
is the Lp space) then
ρ
g
(
X
)
=
−
∫
0
1
F
−
X
−
1
(
p
)
d
g
~
(
p
)
=
∫
−
∞
0
g
~
(
F
−
X
(
x
)
)
d
x
−
∫
0
∞
g
(
1
−
F
−
X
(
x
)
)
d
x
where
F
−
X
is the cumulative distribution function for
−
X
and
g
~
is the dual distortion function
g
~
(
u
)
=
1
−
g
(
1
−
u
)
.
If
X
≤
0
almost surely then
ρ
g
is given by the Choquet integral, i.e.
ρ
g
(
X
)
=
−
∫
0
∞
g
(
1
−
F
−
X
(
x
)
)
d
x
.
Equivalently,
ρ
g
(
X
)
=
E
Q
[
−
X
]
such that
Q
is the probability measure generated by
g
, i.e. for any
A
∈
F
the sigma-algebra then
Q
(
A
)
=
g
(
P
(
A
)
)
.
In addition to the properties of general risk measures, distortion risk measures also have:
- Law invariant: If the distribution of
X
and
Y
are the same then
ρ
g
(
X
)
=
ρ
g
(
Y
)
.
- Monotone with respect to first order stochastic dominance.
- If
g
is a concave distortion function, then
ρ
g
is monotone with respect to second order stochastic dominance.
-
g
is a concave distortion function if and only if
ρ
g
is a coherent risk measure.
Value at risk is a distortion risk measure with associated distortion function
g
(
x
)
=
{
0
if
0
≤
x
<
1
−
α
1
if
1
−
α
≤
x
≤
1
.
Conditional value at risk is a distortion risk measure with associated distortion function
g
(
x
)
=
{
x
1
−
α
if
0
≤
x
<
1
−
α
1
if
1
−
α
≤
x
≤
1
.
The negative expectation is a distortion risk measure with associated distortion function
g
(
x
)
=
x
.
