Matrix Representations of Life Insurance Payments
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.