Life Annuity
The actuarial present value of a life annuity of 1 per year paid continuously can be found in two ways:
Aggregate payment technique (taking the expected value of the total present value):
This is similar to the method for a life insurance policy. This time the random variable Y is the total present value random variable of an annuity of 1 per year, issued to a life aged x, paid continuously as long as the person is alive, and is given by:
where T=T(x) is the future lifetime random variable for a person age x. The expected value of Y is:
Current payment technique (taking the total present value of the function of time representing the expected values of payments):
where F(t) is the cumulative distribution function of the random variable T.
The equivalence follows also from integration by parts.
In practice life annuities are not paid continuously. If the payments are made at the end of each period the actuarial present value is given by
Keeping the total payment per year equal to 1, the longer the period, the smaller the present value is due to two effects:
- The payments are made on average half a period later than in the continuous case.
- There is no proportional payment for the time in the period of death, i.e. a "loss" of payment for on average half a period.
Conversely, for contracts costing an equal lumpsum and having the same internal rate of return, the longer the period between payments, the larger the total payment per year.
Read more about this topic: Actuarial Present Value
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