  Phase-type (PH) distributions are a popular tool for the analysis of
univariate risks in numerous actuarial applications. Their multivariate
counterparts (MPH$^\ast$), however, have not seen such a proliferation, due to
lack of explicit formulas and complicated estimation procedures. A simple
construction of multivariate phase-type distributions -- mPH -- is proposed for
the parametric description of multivariate risks, leading to models of
considerable probabilistic flexibility and statistical tractability. The main
idea is to start different Markov processes at the same state, and allow them
to evolve independently thereafter, leading to dependent absorption times. By
dimension augmentation arguments, this construction can be cast into the
umbrella of MPH$^\ast$ class, but enjoys explicit formulas which the general
specification lacks, including common measures of dependence. Moreover, it is
shown that the class is still rich enough to be dense on the set of
multivariate risks supported on the positive orthant, and it is the smallest
known sub-class to have this property. In particular, the latter result
provides a new short proof of the denseness of the MPH$^\ast$ class. In
practice this means that the mPH class allows for modeling of bivariate risks
with any given correlation or copula. We derive an EM algorithm for its
statistical estimation, and illustrate it on bivariate insurance data.
Extensions to more general settings are outlined.
