Mathematical Derivations of Actuarial Present Value for the Fully Continuous Whole Life Assurances from the Theoretical Market Price

Abstract

Many current mortality tables are computationally Makehamised but technically devoid of continuous key life table functions such as  and  because of the underlying sophisticated mortality functions and the complex methods of computation which usually do not incorporate the market price. The computationally advanced Moore’s model, which does not endorse the market pricing mechanisms offers one of the most complex analytical results that seems not user friendly in actuarial literature. As a result of the Moore’s complexity, we offer the Gradshteyn and Ryzhik’s analytic integral as an alternative solution to examine the contingency issues involving the estimation of actuarial present values of the continuous whole life annuities. In this study, the objective are to (i) estimate the parameters of GM (1,2) through the method of equidistant points according to the natural order of human age (ii) Apply the mean value theorem to construct the survival probability function under the framework of policy modifications (iii) Develop a closed form pricing formula for the continuous whole life annuity and continuous whole life insurance.  The Gradshteyn and Ryzhik’s analytic integral has eliminated the Moore’s complexities associated with symbolic computations such that the continuity assumption applied allows us to compute the continuous life annuity with expedience. Computational evidence further shows that the continuous life annuity reduces with age confirming that life annuities is a decreasing function.