Annuities under Changes in Life Tables and Changes in the Interest

Abstract

A known differential equation for the expectancy of life expresses the force of mortality in terms of the expectancy of life and its derivative. It seems a natural equation to start the study of life assurance when considering changes in the expectancy of life. We derive estimates for the change in annuities for life under a change in the rate of interest. Dynamical life tables (DLT) use force of mortality that varies with time. Life insurance plans and pension schemes are recently considering DLT and variable rate of interest. The evaluation of annuities subject to DLT and variable rates of interest is quite complex. However we observe that in some cases this evaluation may turn out to be relatively simple, by using approximations that are based on the estimates that we achieve. Annuities are known to serve as parameters in evaluating assurances, premiums and reserves. In some cases we are able to approximate annuities under changes in the rate of interest and in the force of mortality. One can express changes in premiums and reserves in terms of the estimates under the changes in life expectancy and the rate of interest. One can use similar ideas to estimate the risk arising from the difference of the volatility of the premiums and that of the liabilities. This may lead to an immunization policy of investing premiums considering the liabilities.

Keywords: rates of interest, force of mortality, expectancy of life, annuities, the classical values ax , Ax , Px , tV x