  This paper presents a novel axiomatic framework of measuring the joint risk
of a portfolio consisting of several financial positions. From the liquidity
shortfall aspect, we construct a distortion-type risk measure to measure the
joint risk of portfolios, which we referred to as multivariate distortion joint
risk measure, representing the liquidity shortfall caused by the joint risk of
portfolios. After its fundamental properties have been studied, we
axiomatically characterize it by proposing a novel set of axioms. Furthermore,
based on the representations for multivariate distortion joint risk measures,
we also propose a new class of vector-valued multivariate distortion joint risk
measures, as well as with sensible financial interpretation. Their fundamental
properties are also investigated. It turns out that this new class is large
enough, as it can not only induce new vector-valued multivariate risk measures,
but also recover some popular vector-valued multivariate risk measures known in
the literature with alternative financial interpretation. Examples are given to
illustrate the proposed multivariate distortion joint risk measures. This paper
mainly gives some theoretical results, helping one to have an insight look at
the measurement of joint risk of portfolios.
