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 [0,+\infty ]}
![{\displaystyle D:{\mathcal {L}}^{2}\to [0,+\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2eef6157a0b2ee321cd1960b2cb956924d2f9c5d)
- Shift-invariant:
D
(
X
+
r
)
=
D
(
X
)
{\displaystyle D(X+r)=D(X)}
for any 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)}
for any X ∈ L 2 {\displaystyle X\in {\mathcal {L}}^{2}}
and λ > 0 {\displaystyle \lambda >0}
- Sublinearity:
D
(
X
+
Y
)
≤
D
(
X
)
+
D
(
Y
)
{\displaystyle D(X+Y)\leq D(X)+D(Y)}
for any X , Y ∈ L 2 {\displaystyle X,Y\in {\mathcal {L}}^{2}}
- Positivity:
D
(
X
)
>
0
{\displaystyle D(X)>0}
for all nonconstant X, and D ( X ) = 0 {\displaystyle D(X)=0}
for any constant X.[1][2]
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} [X])}
-
R
(
X
)
=
D
(
X
)
−
E
[
X
]
{\displaystyle R(X)=D(X)-\mathbb {E} [X]}
.
![{\displaystyle D(X)=R(X-\mathbb {E} [X])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dca13fe97d3397ecd1204a749f14955602493248)
R is expectation bounded if
R
(
X
)
>
E
[
−
X
]
{\displaystyle R(X)>\mathbb {E} [-X]}
![{\displaystyle R(X)>\mathbb {E} [-X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac1dd23763688cfbf8f867365837fc55a83a6060)
![{\displaystyle R(X)=\mathbb {E} [-X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66dfcd7ddf96882c92d876e3d1602325384365cf)
If
D
(
X
)
<
E
[
X
]
−
e
s
s
inf
X
{\displaystyle D(X)<\mathbb {E} [X]-\operatorname {ess\inf } X}
![{\displaystyle D(X)<\mathbb {E} [X]-\operatorname {ess\inf } X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65d828eaf3256468248d1d97dd86e134790e3856)
Examples
The most well-known examples of risk deviation measures are:[1]
- Standard deviation
σ
(
X
)
=
E
[
(
X
−
E
X
)
2
]
{\displaystyle \sigma (X)={\sqrt {E[(X-EX)^{2}]}}}
;
- 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[(X-EX)_{-}}^{2}]}}}
and σ + ( X ) = E [ ( X − E X ) + 2 ] {\displaystyle \sigma _{+}(X)={\sqrt {{E[(X-EX)_{+}}^{2}]}}}
, where [ X ] − := max { 0 , − X } {\displaystyle [X]_{-}:=\max\{0,-X\}}
and [ X ] + := max { 0 , X } {\displaystyle [X]_{+}:=\max\{0,X\}}
;
- Range-based deviations, for example,
D
(
X
)
=
E
X
−
inf
X
{\displaystyle D(X)=EX-\inf X}
and 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)}
by C V a R α Δ ( X ) ≡ E S α ( X − E X ) {\displaystyle {\rm {CVaR}}_{\alpha }^{\Delta }(X)\equiv ES_{\alpha }(X-EX)}
, where E S α ( X ) {\displaystyle ES_{\alpha }(X)}
is Expected shortfall.
See also
- Unitized risk – Relative measure of dispersion expressed as the ratio of standard deviation to the meanPages displaying short descriptions of redirect targets
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).