From Wikipedia, the free encyclopedia
In
financial mathematics and
economics , 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 .
Mathematical definition
The function
ρ
g
:
L
p
→
R
{\displaystyle \rho _{g}:L^{p}\to \mathbb {R} }
associated with the
distortion function
g
:
0
,
1
→
0
,
1
{\displaystyle g:[0,1]\to [0,1]}
is a distortion risk measure if for any
random variable of gains
X
∈
L
p
{\displaystyle X\in L^{p}}
(where
L
p
{\displaystyle 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
{\displaystyle \rho _{g}(X)=-\int _{0}^{1}F_{-X}^{-1}(p)d{\tilde {g}}(p)=\int _{-\infty }^{0}{\tilde {g}}(F_{-X}(x))dx-\int _{0}^{\infty }g(1-F_{-X}(x))dx}
where
F
−
X
{\displaystyle F_{-X}}
is the cumulative distribution function for
−
X
{\displaystyle -X}
and
g
~
{\displaystyle {\tilde {g}}}
is the dual distortion function
g
~
(
u
)
=
1
−
g
(
1
−
u
)
{\displaystyle {\tilde {g}}(u)=1-g(1-u)}
.
[1]
If
X
≤
0
{\displaystyle X\leq 0}
almost surely then
ρ
g
{\displaystyle \rho _{g}}
is given by the
Choquet integral , i.e.
ρ
g
(
X
)
=
−
∫
0
∞
g
(
1
−
F
−
X
(
x
)
)
d
x
.
{\displaystyle \rho _{g}(X)=-\int _{0}^{\infty }g(1-F_{-X}(x))dx.}
[1]
[2] Equivalently,
ρ
g
(
X
)
=
E
Q
−
X
{\displaystyle \rho _{g}(X)=\mathbb {E} ^{\mathbb {Q} }[-X]}
[2] such that
Q
{\displaystyle \mathbb {Q} }
is the
probability measure generated by
g
{\displaystyle g}
, i.e. for any
A
∈
F
{\displaystyle A\in {\mathcal {F}}}
the
sigma-algebra then
Q
(
A
)
=
g
(
P
(
A
)
)
{\displaystyle \mathbb {Q} (A)=g(\mathbb {P} (A))}
.
[3]
Properties
In addition to the properties of general risk measures, distortion risk measures also have:
Law invariant : If the distribution of
X
{\displaystyle X}
and
Y
{\displaystyle Y}
are the same then
ρ
g
(
X
)
=
ρ
g
(
Y
)
{\displaystyle \rho _{g}(X)=\rho _{g}(Y)}
.
Monotone with respect to first order
stochastic dominance .
If
g
{\displaystyle g}
is a
concave distortion function, then
ρ
g
{\displaystyle \rho _{g}}
is monotone with respect to second order stochastic dominance.
g
{\displaystyle g}
is a
concave distortion function if and only if
ρ
g
{\displaystyle \rho _{g}}
is a
coherent risk measure .
[1]
[2]
Examples
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
.
{\displaystyle g(x)={\begin{cases}0&{\text{if }}0\leq x<1-\alpha \\1&{\text{if }}1-\alpha \leq x\leq 1\end{cases}}.}
[2]
[3]
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
.
{\displaystyle g(x)={\begin{cases}{\frac {x}{1-\alpha }}&{\text{if }}0\leq x<1-\alpha \\1&{\text{if }}1-\alpha \leq x\leq 1\end{cases}}.}
[2]
[3]
The negative
expectation is a distortion risk measure with associated distortion function
g
(
x
)
=
x
{\displaystyle g(x)=x}
.
[1]
See also
References
^
a
b
c
d Sereda, E. N.; Bronshtein, E. M.; Rachev, S. T.; Fabozzi, F. J.; Sun, W.; Stoyanov, S. V. (2010). "Distortion Risk Measures in Portfolio Optimization". Handbook of Portfolio Construction . p. 649.
CiteSeerX
10.1.1.316.1053 .
doi :
10.1007/978-0-387-77439-8_25 .
ISBN
978-0-387-77438-1 .
^
a
b
c
d
e Julia L. Wirch; Mary R. Hardy.
"Distortion Risk Measures: Coherence and Stochastic Dominance" (PDF) . Archived from
the original (PDF) on July 5, 2016. Retrieved March 10, 2012 .
^
a
b
c Balbás, A.; Garrido, J.; Mayoral, S. (2008). "Properties of Distortion Risk Measures". Methodology and Computing in Applied Probability . 11 (3): 385.
doi :
10.1007/s11009-008-9089-z .
hdl :
10016/14071 .
S2CID
53327887 .
Wu, Xianyi; Xian Zhou (April 7, 2006). "A new characterization of distortion premiums via countable additivity for comonotonic risks". Insurance: Mathematics and Economics . 38 (2): 324–334.
doi :
10.1016/j.insmatheco.2005.09.002 .