TY - JOUR
T1 - Matrix representations of life insurance payments
AU - Bladt, Mogens
AU - Asmussen, Søren
AU - Steffensen, Mogens
PY - 2020
Y1 - 2020
N2 - 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 methods for finding the reserve and higher order moments involve the solution of certain differential equations (Thiele and Hattendorff, respectively). In this paper we present 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 immediately from the matrix formulae. As a main application, methods for obtaining distributions 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.
AB - 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 methods for finding the reserve and higher order moments involve the solution of certain differential equations (Thiele and Hattendorff, respectively). In this paper we present 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 immediately from the matrix formulae. As a main application, methods for obtaining distributions 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.
KW - Life-insurance
KW - Markov reward processes
KW - Moments
KW - Orthogonal polynomials
KW - Product integral
UR - http://www.scopus.com/inward/record.url?scp=85077715096&partnerID=8YFLogxK
U2 - 10.1007/s13385-019-00222-0
DO - 10.1007/s13385-019-00222-0
M3 - Journal article
AN - SCOPUS:85077715096
VL - 10
SP - 29
EP - 67
JO - European Actuarial Journal
JF - European Actuarial Journal
SN - 2190-9733
IS - 1
ER -