Mathematical and Statistical Methods for Actuarial Sciences and Finance

68 M. Cardin and E. Pagani

We recall here that if 0≤xi<VaRXi,orFXi<pi,thenFX<mini{pi}, while
ifxi ≥VaRXiorFXi ≥ pithenFX ≥maxi

{∑n

i= 1 pi−(n−^1 ),^0

}

.Onlyin

the bivariate case do we then know thatSX=FX−^1 +SX^1 +SX^2. Therefore if
FX≤mini{pi},alsoFX≤ 1 −

∏n
i= 1 (^1 −pi). This lets us consider the bounds for
the joint distribution, not just for the marginals. Finally we can present an interesting
result regarding the representation of subadditive distorted risk measures through
convex combinations of Conditional Values at Risk.

Theorem 3.LetX∈X. Consider a subadditive multivariate distortion in the form
Rf[X]=

∫

Rn+[1−f

(

FX(x)

)

]dx. Then there exists a probability measureμon

[0,1]such that: Rf[X]=

∫ 1

0 CV aR[X;p]dμ(p).

Proof.The multivariate distorted measureRf[X]=

∫

Rn+[1−f

(

FX(x)

)

]dxis sub-

additive if fis a convex, increasing function such that:f :[0,1]→[0,1] with
f( 0 )=0andf( 1 )=1. Letp= 1 −

∏n
i= 1 (^1 −pi), then a probability measure
μ(p)exists such that this functionfcan be represented as:f(u)=

∫ 1

0

(u−p)+
( 1 −p)dμ(p)
withp∈[0,1]. Then,∀X∈X, we can write

Rf[X]=

∫

Rn+

[1−f

(

FX(x)

)

]dx

=

∫

Rn+

[1−

∫ 1

0

(

FX(x)− 1 +

∏n

i= 1 (^1 −pi)

)

∏ +

n

i= 1 (^1 −pi)

]dμ(p)dx

=

∫

Rn+

dx

∫ 1

0

[1−

(

FX(x)− 1 +

∏n

i= 1 (^1 −pi)

)

∏ +

n

i= 1 (^1 −pi)

]dμ(p)

=

∫ 1

0

dμ(p)

∫

Rn+

[1−

(

FX(x)− 1 +

∏n

i= 1 (^1 −pi)

)

∏ +

n

i= 1 (^1 −pi)

]dx

=

∫ 1

0

CV aR[X;p]dμ(p). 

Since not every result about stochastic dominance works in a multivariate setting, we
restrict our attention to the bivariate one. However, this is interesting because it takes
into consideration the riskiness not only of the marginal distributions, but also of the
joint distribution, tracing out a course of action to multivariate generalisations. It is
worth noting that this procedure has something to do with concordance measures (or
measures of dependence), which we will describe later on.
We propose some observations about VaR and CVaR formulated through distor-
tion functions whenXis a random vector with values inR^2 +;wehave:

VaR[X;p]=VaRX 1 VaRX 2

and

CV aR[X;p]=VaRX 1 VaRX 2 +

∫+∞

VaRX 2

∫+∞

VaRX 1

SX(x 1 ,x 2 )

( 1 −p 1 )( 1 −p 2 )

dx 1 dx 2.