Using Phase-Type distribution to determine Actuarial Functions for Whole Life Insurance Policies

Abstract

Background The world life expectancy has had a continual increase for both male and female gender. This increasing trend is expected to continue, hence, the need to have a model that does not have a maximum attainable age. The Lee-carter model suggests that there is no limiting age, however, this model is not su cient for computation of actuarial functions . On the other hand, the De Moivre model which was purposely built for computation of annuity functions, and possibly other actuarial functions, requires a limiting age. There is therefore need to price life contracts by not constraining the future life time of policy holders to a maximum attainable age. Methodology This study applies the phase-type model on AM92 data to compute actuarial functions for whole life insurance policies not limiting the future lifetime of individuals to a maximum attainable age. Assurances and annuities were determined by induction of the Laplace function, moment and probability generating functions of the phase-type distribution. Results and Conclusion The phase-type distribution of Coxian nature is applicable for computation of functions. The one absorbing state is death and ages form the phases of the distribution. A life in state (i) transits to the next phase with rate l while it dies with rate qi =a+bic, where, a is death parameter due to accident while bic is due to biological aging of the life. The parameters l;a;b and c were estimated from the Nelson-Mead algorithm in that the aim was to minimize the mean squared error of the survival function. Results showed that there is small margin of deviation between premiums computed directly from life table functions and those computed by applying a Phase-Type distribution. Also, there is a large underestimation of assurances and overestimation of annuities between the ages 40 and 90. However, there are small deviations for whole life net premiums