The actuarial present value (APV) is the expected value of the present value of a contingent cash flow stream (i.e. a series of payments which may or may not be made). Actuarial present values are typically calculated for the benefit-payment or series of payments associated with life insurance and life annuities. The probability of a future payment is based on assumptions about the person's future mortality which is typically estimated using a life table. Life insurance Whole life insurance pays a pre-determined benefit either at or soon after the insured's death. The symbol (x) is used to denote "a life aged x" where x is a non-random parameter that is assumed to be greater than zero. The actuarial present value of one unit of whole life insurance issued to (x) is denoted by the symbol or in actuarial notation. Let G>0 (the "age at death") be the random variable that models the age at which an individual, such as (x), will die. And let T (the future lifetime random variable) be the time elapsed between age-x and whatever age (x) is at the time the benefit is paid (even though (x) is most likely dead at that time). Since T is a function of G and x we will write T=T(G,x). Finally, let Z be the present value random variable of a whole life insurance benefit of 1 payable at time T. Then: where i is the effective annual interest rate and δ is the equivalent force of interest. To determine the actuarial present value of the benefit we need to calculate the expected value of this random variable Z. Suppose the death benefit is payable at the end of year of death. Then T(G, x) := ceiling(G - x) is the number of "whole years" (rounded upwards) lived by (x) beyond age x, so that the actuarial present value of one unit of insurance is given by: where is the probability that (x) survives to age x+t, and is the probability that (x+t) dies within one year. If the benefit is payable at the moment of death, then T(G,x): = G - x and the actuarial present value of one unit of whole life insurance is calculated as where is the probability density function of T, is the probability of a life age surviving to age and denotes force of mortality at time for a life aged . The