64 M. Cardin and E. Pagani
A risk measure, or a premium principle, is the functionalR:X→R ̃,whereX
is a set of non-negative random vectors andR ̃is the extended real line.
In what follows we present some desirable properties P for risk measures, that are
our proposal to generalise the well known properties for the scalar case:
1.Expectation boundedness:R[X]≥E[X 1 ...Xn]∀X.
2.Non-excessive loading:R[X]≤supω∈ {|X 1 (ω)|,...,|Xn(ω)|}.
3.Translation invariance:R[X+a]=R[X]+ ̄a ∀X,∀a∈Rn,whereais a
vector of sure initial amounts anda ̄is the componentwise product of the elements
ofa.
4.Positive homogeneity of order n:R[cX]=cnR[X] ∀X,∀c≥0.
5.Monotonicity:R[X]≤R[Y] ∀X,Ysuch thatXYin some stochastic sense.
6.Constancy:R[b]=b ̄ ∀b∈Rn. A special case isR[ 0 ]=0, which is called
normalisation property.
7.Subadditivity:R[X+Y]≤R[X]+R[Y] ∀X,Y, which reflects the idea that
risk can be reduced by diversification.
8.Convexity:R[λX+( 1 −λ)Y]≤λR[X]+( 1 −λ)R[Y], ∀X,Yandλ∈
[0,1]; this property implies diversification effects as subadditivity does.
We recall here also some notations about stochastic orderings for multivariate random
variables:XSDYindicates the usual stochastic dominance,XUOYindicates the
upper orthant order,XLOYindicates the lower orthant order,XCYindicates
the concordance order andX SMYindicates the supermodular order. For the
definitions, look them up in, for instance, [5].
Let us now characterise another formulation for stop-loss transform in the multi-
variate setting.
Definition 1.The product stop-loss transform of a randomvectorX∈Xis defined
byπX(t)=E
[
(X 1 −t 1 )+...(Xn−tn)+]
∀t∈Rn.As in the univariate case, we can use this instrument to derive a multivariate stochastic
order:
Definition 2.LetX,Y∈Xbe two randomvectors. We say thatXprecedesYin the
multivariate product stop-loss order
(
XSLnY)
if it holds:πX(t)≤πY(t) ∀t∈Rn.It could be interesting to give some extensions to the theory of risk in the multivariate
case, but sometimes it is not possible and we will be satisfied if the generalisation
works at least in two dimensions. As is well known, different notions are equivalent
in the bivariate case for risks with the same univariate marginal distribution [11], but
this is no longer true forn-variate risks withn≥3[8].