70 M. Cardin and E. Pagani
Now we consider the dihedral groupD 4 of the symmetries on the square [0,1]^2 .We
haveD 4 ={e,r,r^2 ,r^3 ,h,hr,hr^2 ,hr^3 }whereeis the identity,his the reflection
aboutx=^12 andris a 90◦counterclockwise rotation.
A measureμon [0,1]^2 is said to beD 4 -invariant if its value for any Borel set
Aof [0,1]^2 is invariant with respect to the symmetries of the unit square that is
μ(A)=μ(d(A)).
Proposition 2.Ifμis a bounded D 4 -invariant measure on[0,1]^2 ,thereexistα,β∈R
such that the function defined by
ρ((X 1 ,X 2 ))=α∫
[0,1]^2F(X^1 ,X^2 )(x 1 ,x 2 )dμ(FX^1 (x 1 ),FX^2 (x 2 ))−βis a concordance measure.
Proof.A measure of concordance associated to a continuous bivariate random vector
depends only on the copula associated to the vector since a measure of concordance
is invariant under invariant increasing transformation of the random variables. So the
result follows from Theorem 3.1 of [6].
5 A vector-valued measure
In Definition 1 we have introduced the concept of product stop-loss transform for
random vectors. We use this approach to give a definition for a new measure that we
call Product Stop-loss Premium.
Definition 8.Consider a non-negative bivariate randomvectorXand calculate the
Value at Risk of its single components. Product Stop-loss Premium (PSP) is defined
as follows: P S P[X;p]=E
[(
X 1 −VaRX 1)
+(
X 2 −VaRX 2)
+]
.
Of course this definition could be extended also in a general case, writing:
PSP[X;p]=E[(
X 1 −VaRX 1)
+...
(
Xn−VaRXn)
+]
,
but some properties will be different, because not everything stated for the bivariate
case works in the multivariate one.
Our aim is to give a multivariate measure that can detect the joint tail risk of the
distribution. In doing this we also have a representation of the marginal risks and
thus the result is a measure that describes the joint and marginal risk in a simple and
intuitive manner.
We examine in particular the caseX 1 >VaRX 1 andX 2 >VaRX 2 simultane-
ously, since large and small values will tend to be more often associated under the
distribution that dominates the other one.
Random variables are concordant if they tend to be all large together or small
together and in this case we have a measure with non-trivial values when the vari-
ables exceed given thresholds together and are not constant, otherwise we have
PSP[X;p]=0.