In investment, an annuity is a series of payments made at equal intervals.
Annuities can be classified by the frequency of payment dates.
The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other regular interval of time.
A common example is a life annuity, which is paid over the remaining lifetime of the annuitant.
Certain and life annuities are guaranteed to be paid for a number of years and then become contingent on the annuitant being alive.
Valuation of annuities certain may be calculated using formulas depending on the timing of payments.
If the payments are made at the end of the time periods, so that interest is accumulated before the payment, the annuity is called an annuity-immediate, or ordinary annuity.
Mortgage payments are annuity-immediate, interest is earned before being paid.
Annuity due refers to a series of equal payments made at the same interval at the beginning of each period.
The present value of an annuity is the value of a stream of payments, discounted by the interest rate to account for the fact that payments are being made at various moments in the future.
is: In practice, often loans are stated per annum while interest is compounded and payments are made monthly.
The future value of an annuity is the accumulated amount, including payments and interest, of a stream of payments made to an interest-bearing account.
For an annuity-immediate, it is the value immediately after the n-th payment.
is: Example: The present value of a 5-year annuity with a nominal annual interest rate of 12% and monthly payments of $100 is: The rent is understood as either the amount paid at the end of each period in return for an amount PV borrowed at time zero, the principal of the loan, or the amount paid out by an interest-bearing account at the end of each period when the amount PV is invested at time zero, and the account becomes zero with the n-th withdrawal.
Similarly, we can prove the formula for the future value.
Therefore, An annuity-due is an annuity whose payments are made at the beginning of each period.
[3] Deposits in savings, rent or lease payments, and insurance premiums are examples of annuities due.
Each annuity payment is allowed to compound for one extra period.
The future and present values for annuities due are related since: Example: The final value of a 7-year annuity-due with a nominal annual interest rate of 9% and monthly payments of $100 can be calculated by: In Excel, the PV and FV functions take on optional fifth argument which selects from annuity-immediate or annuity-due.
Thus we have: A perpetuity is an annuity for which the payments continue forever.
Observe that Therefore a perpetuity has a finite present value when there is a non-zero discount rate.
Life tables are used to calculate the probability that the annuitant lives to each future payment period.
Also, this can be thought of as the present value of the remaining payments See also fixed rate mortgage.