Life Annuities: Whole Life, Temporary, and Deferred

Whole Life Annuities

A whole life annuity-due pays 1 at the beginning of each year while (x) survives. The APV is a-ddot_x = sum from k=0 to infinity of v^k * k_p_x. A whole life annuity-immediate pays at year end: a_x = a-ddot_x minus 1. The continuous whole life annuity pays at rate 1 per year: a-bar_x = integral from 0 to infinity of v^t * t_p_x dt. The fundamental relationship connecting insurance and annuities is A_x = 1 minus d * a-ddot_x (discrete) and A-bar_x = 1 minus delta * a-bar_x (continuous), where d = i/(1+i) and delta = ln(1+i).

Temporary and Deferred Annuities

An n-year temporary annuity-due pays 1 at the start of each year for at most n years while (x) survives: a-ddot_{x:n} = sum from k=0 to n-1 of v^k * k_p_x. An n-year deferred annuity-due begins payments at time n: n|a-ddot_x = n_E_x * a-ddot_{x+n}. The decomposition a-ddot_x = a-ddot_{x:n} + n|a-ddot_x partitions whole life into temporary plus deferred components. Exam LTAM problems require fluency with these formulas and the ability to derive one annuity value from others using relationships.