66 M. Cardin and E. Pagani
concavity: ifgis a concave function, then we have thatg(a+c)−g(a)≥g(b+c)−
g(b)witha≤bandc≥0. We apply this definition pointwise toSX≤SYwith
SX+Y≥0. P8 is obvious from properties P4, P5 and P7.
In the multivariate case the equalityFX = 1 −SXdoes not hold and thus, the
relation
∫
Rn+g(
SX(x))
dx=∫
Rn+[1−f(
FX(x))
]dxis not in general true with
f:[0,1]→[0,1], increasing function.
Moreover, the duality relationship between the functionsfandgdoes not hold,
thus, in general, the equationg(x)= 1 −f( 1 −x)is not true. Applying the concept
of distortion of either the survival function or the distribution function, the relationship
betweenfandgno longer holds.
Therefore we can observe the differences in the two different approaches.
Definition 4.Given a distortion function f :[0,1]→[0,1], increasing and such
that f( 0 )= 0 and f( 1 )= 1 , a vector distorted risk measure is the functional:
Rf[X]=∫
Rn+[1−f(
FX(x))
]dx.Now we have subadditivity with a convex functionfand this leads to the convexity
of the measureRf.
Remembering that a distortion is a univariate function even when we deal with
random vectors and multivariate distributions, we can also define vector Values at
Risk (VaR) and vector Conditional Values at Risk (CVaR), using slight alterations of
the usual distortions for VaR and CVaR respectively, and composing these with the
multivariate tail distributions or the distribution functions.
Definition 5.LetXbe a randomvector that takes on values inRn+.VectorVaRisthe
distorted measure V a R[X;p] =
∫+∞
0 ...
∫+∞
0 g(
SX(x))
dx 1 ...dxn,expressed
using the distortion
g(
SX(x))
=
{
00 ≤SXi(xi)≤ 1 −pi
11 −pi<SXi(xi)≤ 1.
If we want to give to this formulation a more explicit form we can consider the
componentwise order for whichx>VaR[X;p] stands forxi>VaR[Xi;p]∀i=
1 ,...,nor more lightlyxi>VaRXiand we can rewrite the distortion as:
g(
SX(x))
=
{
0 xi≥VaRXi
10 ≤xi<VaRXi.
to obtainVaR[X;p]=
∫VaRXn
0 ...∫VaRX 1
0 1 dx^1 ...dxn=VaRX 1 ...VaRXn.Ob-
viously this result suggests that considering a componentwise order is similar to con-
sidering an independency between the components of the random vector. Actually
we are considering only the case in which the components are concordant.