Actuarial Calculations Using a Markov Model

This paper develops a general approach to actuarial calculations in applications which can be modeled as multi-state processes. Such situations arise when beneets are payable upon a change in the status of the insured, or while the insured maintains a given status. Examples include life insurance, annuities, pensions, disability income insurance and certain types of long-term care insurance. The method is based on convenient matrix results which are available when we assume a continuous-time Markov model with constant forces of transition. In this case probabilities are easily obtained regardless of the number of states. This is of considerable beneet as it allows us to deal with very complicated actuarial problems. Section 1 provides some background on the kinds of problems for which the approach is suitable. The basic properties of the Markov process are presented in Section 2. We consider some useful results which hold under the assumption of constant forces of transition, and explain how the results can be used in the case of piecewise constant forces. In Section 3, we address the situation in which the Markov assumption is inappropriate. Here we nd that, rather than using a more general semi-Markov model, we can reeect duration dependence by increasing the number of states in the model. This is justiied by a limiting result and demonstrated by an example which applies the approach to select and ultimate mortality.