Actuarial present value

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" and the actuarial present value of one unit of whole life insurance issued to (x) is denoted by the symbol $\,A_{x}\!$ or $\,{\overline {A}}_{x}\!$ in actuarial notation. Let G (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:

$\,Z=v^{T}=(1+i)^{-T}$

where i is the effective annual interest rate.

To determine the actuarial present value of the benefit we need to calculate the expected value $\,E(Z)$ of this random variable Z. Suppose the death benefit is payable at the end of year of death. Then the actuarial present value of one unit of insurance is

$A_{x}=E(Z)=E(v^{T})=\sum _{t=0}^{\infty }v^{t+1}Pr[T(G,x)=t]=\sum _{t=0}^{\infty }v^{t+1}Pr[t<G-x\leq t+1\mid G>x]=\sum _{t=1}^{\infty }(1+i)^{-(t+1)}{}_{t}p_{x}\cdot q_{x+t},$

where T(G, x) = ceiling(G - x) is the number of "whole years" lived by (x) beyond age x, ${}_{t}p_{x}$ is the probability that (x) survives to age x+t, and $\,q_{x+t}$ is the probability that (x+t) dies within one year.

If the benefit is payable at the moment of death, then the actuarial present value of one unit of whole life insurance is calculated as

$\,{\overline {A}}_{x}\!=E(v^{T})=\int _{0}^{\infty }v^{t}f_{T}(t)\,dt=\int _{0}^{\infty }v^{t}\,_{t}p_{x}\mu _{x+t}\,dt,$

where $f_{T}$ is the probability density function of T := G - x, $\,_{t}p_{x}\!$ is the probability of a life age $x$ surviving to age $x+t$ and $\mu _{x+t}$ denotes force of mortality at time $x+t$ for a life aged $x$.

The actuarial present value of one unit of an n-year term insurance policy payable at the moment of death can be found similarly by integrating from 0 to n.

The actuarial present value of an n year pure endowment insurance benefit of 1 payable after n years if alive, can be found as

$\,_{n}E_{x}=P(T>n)v^{n}=\,_{n}p_{x}v^{n}.$

In practice the information available about the random variable T may be drawn from life tables, which give figures by year. For example, a three year term life insurance of $100,000 payable at the end of year of death has actuarial present value

$100,000\,A_{{\stackrel {1}{x}}:{{\overline {3}}|}}=100,000\sum _{t=0}^{2}v^{t+1}Pr[T(G,x)=t].$

Suppose for example, that the probability of survival is 0.9 for all ages. Then

$Pr[T(G,x)=0]=0.1,\quad Pr[T(G,x)=1]=0.9(0.1)=0.09,\quad Pr[T(G,x)=2]=0.9^{2}(0.1)=0.081,$

and at interest rate 6% the actuarial present value of one unit of the three year term insurance is

$\,A_{{\stackrel {1}{x}}:{{\overline {3}}|}}=0.1(1.06)^{-1}+0.09(1.06)^{-2}+0.081(1.06)^{-3}=0.24244846,$

so the actuarial present value of the $100,000 insurance is $24,244.85.

In practice the benefit may be payable at the end of a shorter period than a year, which requires an adjustment of the formula.