Whole Life Insurance: Continuous and Discrete Models
Compare continuous and discrete whole life insurance models and their APV calculations for Exam LTAM.
Continuous Whole Life Insurance
A continuous whole life insurance pays a benefit of 1 at the exact moment of death. The present value random variable is Z = v^{T_x} where T_x is the continuous future lifetime. The APV is A-bar_x = integral from 0 to infinity of v^t * t_p_x * mu_{x+t} dt. Under constant force of mortality mu, A-bar_x = mu/(mu + delta) where delta = ln(1+i). The variance of Z is (2*A-bar_x) minus (A-bar_x)^2, where the superscript 2 denotes evaluation at twice the force of interest.
Discrete Whole Life Insurance
A discrete whole life insurance pays 1 at the end of the year of death. The present value random variable is Z = v^{K_x + 1} where K_x is the curtate future lifetime. The APV is A_x = sum over k=0 to infinity of v^{k+1} * k_p_x * q_{x+k}. The recursion A_x = v*q_x + v*p_x*A_{x+1} is useful for computation. The 1/mthly case pays at the end of the 1/m-th of a year of death. Exam LTAM tests conversions between continuous, annual, and 1/mthly models using relationships like A_x approximately equals i/delta times A-bar_x.