In financial mathematics , a deviation risk measure is a function to quantify financial risk (and not necessarily downside risk ) in a different method than a general risk measure . Deviation risk measures generalize the concept of standard deviation .
Mathematical definition
A function
D
:
L
2
→
[
0
,
+
∞
]
{\displaystyle D:{\mathcal {L}}^{2}\to }
L
2
{\displaystyle {\mathcal {L}}^{2}}
L2 space of random variables (random portfolio returns ), is a deviation risk measure if
Shift-invariant:
D
(
X
+
r
)
=
D
(
X
)
{\displaystyle D(X+r)=D(X)}
r
∈
R
{\displaystyle r\in \mathbb {R} }
Normalization:
D
(
0
)
=
0
{\displaystyle D(0)=0}
Positively homogeneous:
D
(
λ
X
)
=
λ
D
(
X
)
{\displaystyle D(\lambda X)=\lambda D(X)}
X
∈
L
2
{\displaystyle X\in {\mathcal {L}}^{2}}
λ
>
0
{\displaystyle \lambda >0}
Sublinearity:
D
(
X
+
Y
)
≤
D
(
X
)
+
D
(
Y
)
{\displaystyle D(X+Y)\leq D(X)+D(Y)}
X
,
Y
∈
L
2
{\displaystyle X,Y\in {\mathcal {L}}^{2}}
Positivity:
D
(
X
)
>
0
{\displaystyle D(X)>0}
X , and
D
(
X
)
=
0
{\displaystyle D(X)=0}
X . Relation to risk measure
There is a one-to-one relationship between a deviation risk measure D and an expectation-bounded risk measure R where for any
X
∈
L
2
{\displaystyle X\in {\mathcal {L}}^{2}}
D
(
X
)
=
R
(
X
−
E
[
X
]
)
{\displaystyle D(X)=R(X-\mathbb {E} )}
R
(
X
)
=
D
(
X
)
−
E
[
X
]
{\displaystyle R(X)=D(X)-\mathbb {E} }
R is expectation bounded if
R
(
X
)
>
E
[
−
X
]
{\displaystyle R(X)>\mathbb {E} }
X and
R
(
X
)
=
E
[
−
X
]
{\displaystyle R(X)=\mathbb {E} }
X .
If
D
(
X
)
<
E
[
X
]
−
e
s
s
inf
X
{\displaystyle D(X)<\mathbb {E} -\operatorname {ess\inf } X}
X (where
e
s
s
inf
{\displaystyle \operatorname {ess\inf } }
essential infimum ), then there is a relationship between D and a coherent risk measure .
Examples
The most well-known examples of risk deviation measures are:
Standard deviation
σ
(
X
)
=
E
[
(
X
−
E
X
)
2
]
{\displaystyle \sigma (X)={\sqrt {E}}}
Average absolute deviation
M
A
D
(
X
)
=
E
(
|
X
−
E
X
|
)
{\displaystyle MAD(X)=E(|X-EX|)}
Lower and upper semi-deviations
σ
−
(
X
)
=
E
[
(
X
−
E
X
)
−
2
]
{\displaystyle \sigma _{-}(X)={\sqrt {{E}}}
σ
+
(
X
)
=
E
[
(
X
−
E
X
)
+
2
]
{\displaystyle \sigma _{+}(X)={\sqrt {{E}}}
[
X
]
−
:=
max
{
0
,
−
X
}
{\displaystyle _{-}:=\max\{0,-X\}}
[
X
]
+
:=
max
{
0
,
X
}
{\displaystyle _{+}:=\max\{0,X\}}
Range-based deviations, for example,
D
(
X
)
=
E
X
−
inf
X
{\displaystyle D(X)=EX-\inf X}
D
(
X
)
=
sup
X
−
inf
X
{\displaystyle D(X)=\sup X-\inf X}
Conditional value-at-risk (CVaR) deviation, defined for any
α
∈
(
0
,
1
)
{\displaystyle \alpha \in (0,1)}
C
V
a
R
α
Δ
(
X
)
≡
E
S
α
(
X
−
E
X
)
{\displaystyle {\rm {CVaR}}_{\alpha }^{\Delta }(X)\equiv ES_{\alpha }(X-EX)}
E
S
α
(
X
)
{\displaystyle ES_{\alpha }(X)}
Expected shortfall . See also
References
^ Rockafellar, Tyrrell; Uryasev, Stanislav; Zabarankin, Michael (2002). "Deviation Measures in Risk Analysis and Optimization". SSRN 365640 . {{cite journal }}: Cite journal requires |journal= (help )
Cheng, Siwei; Liu, Yanhui; Wang, Shouyang (2004). "Progress in Risk Measurement". Advanced Modelling and Optimization . 6 (1).
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