Some classes of multivariate risk measures 69Under independence hypothesis we can rewrite CVaR in this manner:
CV aR[X;p]=VaRX 1 VaRX 2+1
( 1 −p 1 )( 1 −p 2 )∫+∞
VaRX 2∫+∞
VaRX 1SX^1 (x 1 )SX^2 (x 2 )dx 1 dx 2.Then we consider
∫+∞VaRX 2∫+∞
VaRX 1SX^1 (x 1 )SX^2 (x 2 )dx 1 dx 2 =( 1 −p 1 )( 1 −p 2 )VaRX 1 VaRX 2 +( 1 −p 1 )VaRX 1∫+∞
VaRX 2x 2 dSX^2 (x 2 )+( 1 −p 2 )VaRX 2∫+∞
VaRX 1x 1 dSX^1 (x 1 )+∫+∞
VaRX 1x 1 dSX^1 (x 1 )∫+∞
VaRX 2x 2 dSX^2 (x 2 ),which leads, with the first part, to:
CV aR[X;p]= 2 VaRX 1 VaRX 2 −VaRX 1 E[
X 2 |X 2 >VaRX 2]
−
VaRX 2 E[
X 1 |X 1 >VaRX 1]
+E
[
X 2 |X 2 >VaRX 2]
E
[
X 1 |X 1 >VaRX 1]
.
4 Measures of concordance
Concordance between two random variables arises if large values tend to occur with
large values of the other and small values occur with small values of the other. So
concordance considers nonlinear associations between random variables that corre-
lation might miss. Now, we want to consider the main characteristics a measure of
concordance should have. We restrict our attention to the bivariate case.
In 1984 Scarsini ( [10]) defined a set of axioms that a bivariate dependence or-
dering of distributions should have in order that higher ordering means more positive
concordance.
By ameasure of concordancewe mean a function that attaches to every continuous
bivariate random vector a real numberα(X 1 ,X 2 )satisfying the following properties:
1.− 1 ≤α(X 1 ,X 2 )≤1;
2.α(X 1 ,X 1 )=1;
3.α(X 1 ,−X 1 )=−1;
4.α(−X 1 ,X 2 )=α(X 1 ,−X 2 )=−α(X 1 ,X 2 );
5.α(X 1 ,X 2 )=α(X 2 ,X 1 );
- ifX 1 andX 2 are independent, thenα(X 1 ,X 2 )=0;
- if(X 1 ,X 2 )C(Y 1 ,Y 2 )thenα(X 1 ,X 2 )≤α(Y 1 ,Y 2 )
- if{X}nis a sequence of bivariate random vectors converging in distribution toX,
then limn→∞α(Xn)=α(X).