An annuity is a series of equal cash flows over time that comes at regular intervals. The cash flows must be either all payments or all receipts, consistently occur either at the beginning or the end of the interval and represent one discount period. Payments made at the beginning of the period indicate an "annuity due" which can include rents and insurance payments. Payments at the end of the period indicate an "ordinary annuity" which include mortgage payments, bond payments, etc.

Although loan payments, mortgages and similar financial instruments can be regarded as an annuity, the term is mostly applied from the perspective of being an asset. For example, payments from a lottery or distributions from a lump-sum amount can be considered as an annuity. Annuities can also be an investment used to guarantee a regular income during a retirement.

Calculating annuity payments can come from two perspectives: the future value of an annuity or the present value of an annuity.

Calculating Ordinary Annuity Payments From Future Value

If the desired ending amount is known together with the discount rate and number of periods, the payments can be calculated as follows:

PMT = FV / (((1 + r)^n - 1) / r)

Where:

PMT = Payment amount made at the end of the period

FV = The future value of the annuity (how much the balance will be after all payments have been made)

r = the discount rate

^ = raises the value to the left to an exponential number on the right

n = the number of payments

In this calculation, the present value (PV) is assumed to be zero.

Calculating Ordinary Annuity Payments From Present Value

If the sum of money or balance on hand is known together with the discount rate and the number of periods, the amount of payments to reduce the balance to zero can be calculated as follows:

PMT = PV / ((1-[1 / (1 + r)^n] )/ r)

Where:

PMT = Payment amount made at the end of the period

PV = The present value of the annuity (how much is currently on hand)

r = the discount rate

^ = raises the value to the left to an exponential number on the right

n = the number of payments

In this calculation, the future value (FV) is assumed to be zero.

Calculating Annuity Due Payments From Future Value

Because the payment earns interest for one additional period than the ordinary annuity, the future value should be adjusted as follows:

FV annuity due = FV ordinary annuity X (1+r)

The new value for future value can now be inserted in the original equation to compute the annuity due payments.

Calculating Annuity Due Payments From Present Value

To remove the additional discount period for each payment made on an annuity due, the present value of the annuity must be adjusted as follows:

PV annuity due = PV ordinary annuity X (1+r)

The new value for future value can now be inserted in the original equation to compute the annuity due payments.

Alternate Methods

Because calculating the payments for ordinary annuities and annuities due, a financial calculator such as the HP 10bII can be used to simplify the process. When many calculations must be performed, the process can be expedited through the use of a spreadsheet such as Microsoft Excel which is equipped with time value of money functions.

See the related links below for an annuity calculator for different types of contracts that compute the balance, distributions, or present value using the amounts you specify.

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