In actuarial science, the actuarial present value of a payment or series of payments which are random variables is the expected value of the present value of the payments, or equivalently, the present value of their expected values.
This applies to life insurances, including life annuities. The result depends on age through life tables, and on the interest rate.
The internal rate of return of a contract is the rate of return for which the actuarial present value of all cash flows is zero.
Let be the future lifetime random variable of an individual age x and be the present value random variable of a whole life insurance benefit of 1 payable at the instant of death.
where "i" is the interest rate and δ is the equivalent force of interest.
To calculate the actuarial present value we need to calculate the expected value of this random variable Z. For someone aged "x" this is denoted as in actuarial notation. It can be calculated as
where is the probability density function of "T", is the probability of a life age surviving to age and "μ" denotes force of mortality.
The actuarial present value of an n-year term insurance policy 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
In practice the best information available about the random variable "T" is drawn from life tables, which give figures by year. The actuarial present value of a benefit of 1 payable at the birthday after death would be