  It is well-known that combining life annuities and death benefits introduce
opposite effects in payments with respect to the unsystematic mortality risk on
the lifetime of the insured. In a general multi-state framework with multiple
product types, such joint effects are less trivial. In this paper, we consider
a multivariate payment process in multi-state life insurance, where the
components are defined in terms of the same Markovian state process. The
multivariate present value of future payments is introduced, and we derive
differential equations and product integral representations of its conditional
moments and moment generating function. Special attention is given to pair-wise
covariances between two present values, where results closely connected to
Hattendorff type of results for the variance are derived. Numerical examples
are shown in a disability model and a spouse model to illustrate applicability
of the results on insurance of a single life as well as two dependent lives.
