72 M. Cardin and E. Pagani
We can note that our PSP is not always subadditive; in fact, if we take the non-
negative vectorsX,Y∈X, the following relation is not always satisfied:
E[(
X 1 +Y 1 −VaRX 1 −VaRY 1)
+(
X 2 +Y 2 −VaRX 2 −VaRY 2)
+]
≤
E
[(
X 1 −VaRX 1)
+(
X 2 −VaRX 2)
+]
+E
[(
Y 1 −VaRY 1)
+(
Y 2 −VaRY 2)
+]
.
After verifying all the possible combinations among scenarios
X 1 >VaRX 1 , X 1 <VaRX 1 , X 2 >VaRX 2 , X 2 <VaRX 2 ,
Y 1 >VaRY 1 , Y 1 <VaRY 1 , Y 2 >VaRY 2 , Y 2 <VaRY 2 ,we can conclude that the measure is not subadditive when:
- the sum of the components is concordant and such that:
Xi+Yi>VaRXi+VaRYi∀i= 1 , 2 ,
with discordant components of at most one vector;- the sum of the components is concordant and such that:
Xi+Yi>VaRXi+VaRYi∀i= 1 , 2 ,
with both vectors that have concordant components, but with a different sign: i.e.,
Xi>(<)VaRXi and Yi<(>)VaRYi ∀i;- Xi>VaRXi and Yi>VaRYi∀i simultaneously.
Hence, in these cases, the measure reflects the dependence structure of the vectors
involved.
6 Conclusions
In this paper we have proposed a mathematical framework for the introduction of
multivariate measures of risk. After considering the main properties a vector measure
should have, and recalling some stochastic orders, we have outlined our results on
multivariate risk measures. First of all, we have generalised the theory about dis-
torted risk measures for the multivariate case, giving a representation result for those
measures that are subadditive and defining the vector VaR and CVaR. Then, we have
introduced a new risk measure, called Product Stop-Loss Premium, through its defini-
tion, its main properties and its relationships with CVaR and measures of concordance.
This measure lets us also propose a new stochastic order. We can observe that, in the
literature, there are other attempts to study multivariate risk measures, we cite for
example [1–3, 7] and [9], but they all approach the argument from different points of
view. Indeed, [9] is the first work that deals with multivariate distorted risk measures,
but it represents only an outline for further developments, as we have done in the
present work.
More recently, the study of risk measures has focused on weakening the definition
of convenient properties for risk measures, in order to represent the markets in a more
faithful manner, or on the generalisation of the space that collects the random vectors.