The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time. This usage is most commonly seen in discussions of finance, usually in connection with the valuation of the stream of payments, taking into account time value of money concepts such as interest rate and future value.^{[1]}
Examples of annuities are regular deposits to a savings account, monthly home mortgage payments and monthly insurance payments. Annuities are classified by payment dates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other interval of time.
Contents 
An ordinary annuity (also referred as annuityimmediate) is an annuity whose payments are made at the end of each period (e.g. a month, a year). The values of an ordinary annuity can be calculated through the following:^{[2]}
Let:
Note:
Also let:
Also:
Clearly, in the limit as n increases,
Thus, even an infinite series of finite payments (perpetuity) with a nonzero discount rate has a finite present value.
The next payment is to be paid in one period. Thus, the present value is computed to be:
We notice that the second term is a geometric progression of scale factor 1 and of common ratio . We can write
Finally, after simplifications, we obtain
Similarly, we can prove the formula for the future value. The payment made at the end of the last year would accumulate no interest and the payment made at the end of the first year would accumulate interest for a total of (n−1) years. Therefore,
Hence:
If an annuity is for repaying a debt P with interest, the amount owed after n payments is:
because the scheme is equivalent with lending an amount and putting part of that, an amount , in the bank to grow due to interest. See also fixed rate mortgage.
An annuitydue is an annuity whose payments are made at the beginning of each period.^{[3]} Deposits in savings, rent payments, and insurance premiums are examples of annuities due.
Because each annuity payment is allowed to compound for one extra period, the value of an annuitydue is equal to the value of the corresponding ordinary annuity multiplied by (1+i). Thus, the future value of an annuitydue can be calculated through the formula (variables named as above):^{[4]}
It can also be written as
An annuitydue with n payments is the sum of one annuity payment now and an ordinary annuity with one payment less, and also equal, with a time shift, to an ordinary annuity with one payment more, minus the last payment.
Thus we have:
Annuity due is useful for lease payment calculations 1
