Winner of the GAUSS Prize 2020

A  multi-state  life  insurance  model  is  described  naturally  in  terms  of  the  intensity  matrix of an underlying (time-inhomogeneous) Markov process which specifies the dynamics  for  the  states  of  an  insured  person.  Between  and  at  transitions,  benefits  and  premiums  are  paid,  defining  a  payment  process,  and  the  technical  reserve  is  defined as the present value of all future payments of the contract. Classical meth-ods for finding the reserve and higher order moments involve the solution of certain differential  equations  (Thiele  and  Hattendorff,  respectively).  In  this  paper  we  pre-sent an alternative matrix-oriented approach based on general reward considerations for Markov jump processes. The matrix approach provides a general framework for effortlessly setting up general and even complex multi-state models, where moments of all orders are then expressed explicitly in terms of so-called product integrals of certain matrices. Thiele and Hattendorff type of theorems may be retrieved immedi-ately from the matrix formulae. As a main application, methods for obtaining distri-butions and related properties of interest (e.g. quantiles or survival functions) of the future  payments  are  presented  from  both  a  theoretical  and  practical  point  of  view,  employing Laplace transforms and methods involving orthogonal polynomials.