This paper initiates the extended Makeham’s function deploying rigorous analytic constructions with interesting results for the design of life insurance benefits. The information reinforcing the actuarial valuation of life insurance benefits is dependent on substantiated contingencies and underwriting transactions without reflecting the expected future trends. The objective is to employ refinements to existing methods to capture mortality rate intensities accurately for actuarial valuation. The theoretical underpinnings of instantaneous mortality rate model were explored by conducting deep analytic derivations of model equations through numerical discretisation to examine the degree to which the functional equations impact the mortality-dependant and the actuarial valuation of life insurance benefits. The procedure employed numerical discretisation to prove the benefit function theorem which pricing actuaries depend upon to derive the net premiums. Computational evidence from the results, reveal that the benefit functions exhibit full temporal dynamics of life insurance valuation satisfying the duality principle. The trajectories of both benefit functions also confirm monotonicity and smoothness but the smooth evolution implies actuarial coherence and continuity of the underlying mortality intensity and force of interest. The monotonic behaviour demonstrates the trade-off between longevity risk and mortality risk which are essential to insurance products design. The continuous- discrete relationship demonstrates analytic continuum which serves as iconic bridge between theoretical and applied life insurance practice. Since the trajectories of life annuity and insurance functions both satisfy the duality principle, it is therefore recommended that the law be employed by actuaries when conducting actuarial valuations of life insurance schemes.
Keywords: Contingencies, discretisation, duality, refinement, mortality
