Mathematical and Statistical Methods for Actuarial Sciences and Finance

Some classes of multivariate risk measures 71

It is clear that concordance affects this measure, and in general we know that
concordance behaviour influences risk management of large portfolios of insurance
contracts or financial assets. In these portfolios the main risk is the occurrence of
many joint default events or simultaneous downside evolutions of prices.
PSP for multivariate distributions is interpreted as a measure that can keep the
dependence structure of the components of the random vector considered, when spec-
ified thresholds are exceeded by each component with probabilitypi; but indeed it
is also a measure that can evaluate the joint as well as the marginal risk. In fact, we
have:

PSP[X;p]=E

[(

X 1 −VaRX 1

)

+

(

X 2 −VaRX 2

)

+

]

=

∫+∞

VaRX 2

∫+∞

VaRX 1

SX(x)dx 1 dx 2 −VaRX 2 E

[

X 1 |X 1 >VaRX 1

]

−VaRX 1 E

[

X 2 |X 2 >VaRX 2

]

+VaRX 1 VaRX 2.

Let us denote withCV aR[X;p] the CVaR restricted to the bivariate independent
case, withX 1 >VaRX 1 andX 2 >VaRX 2 ,thenwehave:

CV aR[X;p]=E

[

X 1 |X 1 >VaRX 1

]

E

[

X 2 |X 2 >VaRX 2

]

−

VaRX 1 E

[

X 2 |X 2 >VaRX 2

]

−VaRX 2 E

[

X 1 |X 1 >VaRX 1

]

+VaRX 1 VaRX 2.

We can conclude that these risk measures are the same for bivariate vectors with
independent components, on the condition of these restrictions.
We propose here a way to compare dependence, introducing a stochastic order
based on our PSP measure.

Proposition 3.IfX,Y∈R 2 ,thenXSMY ⇐⇒ PSP[X;p]≤PSP[Y;p]∀p
holds.

Proof.IfXSMYthenE[f(X)]≤E[f(Y)] for every supermodular functionf,
therefore also for the specific supermodular function that defines our PSP and then
followsPSP[X;p]≤PSP[Y;p]. Conversely if

PSP[X;p]≤PSP[Y;p]andX,Y∈R 2 ,

we have ∫
+∞

VaRX 2

∫+∞

VaRX 1

SX(t)dt≤

∫+∞

VaRY 2

∫+∞

VaRY 1

SY(t)dt

withVaRX 2 =VaRY 2 andVaRX 1 =VaRY 1. It follows thatSX(t)≤SY(t),which
leads toXCY. From Theorem 1 followsXSMY. 

Obviously PSP is also consistent with the concordance order.
Another discussed property for risk measures is subadditivity; risk measures that
are subadditive for all possible dependence structures of the vectors do not reflect the
dependence between(X 1 −α 1 )+and(X 2 −α 2 )+.