Mathematical and Statistical Methods for Actuarial Sciences and Finance

Some classes of multivariate risk measures 67

In the same way we can define the vector Conditional Value at Risk:

Definition 6.LetXbe a randomvector with values inRn+. Vector CVaR is the distorted
measure C V a R[X;p]=

∫+∞

0 ...

∫+∞

0 g

(

SX(x)

)

dx 1 ...dxn,expressed using the
distortion:

g

(

SX(x)

)

=

⎧

⎨

⎩

SX(x)

∏n

i= 1 (^1 −pi)

0 ≤SXi(xi)≤ 1 −pi

11 −pi<SXi(xi)≤ 1

.

A more tractable form is given by:

g

(

SX(x)

)

=

⎧

⎨

⎩

SX(x)

∏n

i= 1 (^1 −pi)

xi≥VaRXi

10 ≤xi<VaRXi

,

which allows this formula:

CV aR[X;p]=

∫VaRXn

0

...

∫VaRX

1

0

1 dx 1 ...dxn+

∫+∞

VaRXn

...

∫+∞

VaRX 1

SX(x)

∏n

i= 1 (^1 −pi)

dx 1 ...dxn=

VaR[X;p]+

∫+∞

VaRXn

...

∫+∞

VaRX 1

SX(x)

∏n

i= 1 (^1 −pi)

dx 1 ...dxn.

The second part of the formula is not easy to render explicitly if we do not introduce
an independence hypothesis.
If we follow Definition 4 instead of 3 we can introduce a different formulation for
CVaR, very useful in proving a good result proposed later on.
The increasing convex functionfused in the definition of CVaR is the following:

f

(

FX(x)

)

=

⎧

⎪⎨

⎪⎩

0 FXi(xi)<pi 0 ≤xi<VaRXi

FX(x)− 1 +

∏n

∏ i=^1 (^1 −pi)

n

i= 1 (^1 −pi)

FXi(xi)≥pi xi≥VaRXi

Definition 7.LetXbe a randomvector that takes on values inRn+and f be an
increasing function f:[0,1]→[0,1], such that f( 0 )= 0 and f( 1 )= 1 and defined
as above. The Conditional Value at Risk distorted by such an f is the following:

CV aR[X;p]=

∫ VaRXn

0

...

∫VaRX

1

0

1 dx 1 ...dxn+

∫+∞

VaRXn

...

∫+∞

VaRX 1

[

1 −

FX(x)− 1 +

∏n

∏ i=^1 (^1 −pi)

n

i= 1 (^1 −pi)

]

dx 1 ...dxn=

∫+∞

0

...

∫+∞

0

1 −

[FX(x)− 1 +

∏n

∏ i=^1 (^1 −pi)]+

n

i= 1 (^1 −pi)

dx 1 ...dxn.