# Ordinary Annuity

## Definition

The term ordinary annuity refers to a series of payments, or receipts of money, occurring at consistent intervals of time with an interest charge applied once per interval.  With an ordinary annuity, the payment or receipt of money occurs at the end of each interval.

### Calculation

Present Value of an Ordinary Annuity:

= P x [1 - (1 + i)-N] / i

Future Value of an Ordinary Annuity:

= P x [(1 + i)N - 1] / i

Where:

• P = payment or receipt of money
• N = number of intervals or transactions
• i = interest rate

### Explanation

For the receipt or payment of money to be considered an annuity, it must possess all three of the following attributes:

• The amount of money exchanged in each interval must be the same
• The time intervals must be of the same length
• Interest charges are applied once per interval

An annuity can be as simple as the periodic deposit of money into a savings account, or as complex as a life annuity, which is an insurance product that is used by retirees to provide for a steady stream of income.

With an ordinary annuity, also known as an annuity-immediate, the payment or receipt of money occurs at the end of each time interval.  This differs from an annuity due, which requires the transaction to occur at the beginning of each time interval.  For this reason, the number of compounding intervals for an ordinary annuity will be one less than the total number of transactions.

### Example

On January 1st, Bill made a commitment that he would deposit \$5,000 at the end of each year into his retirement account for the next 20 years.  The rate of interest earned on that money will be 5%.  Bill would like to calculate both the present value and future value of this retirement account balance at the end of 20 years.

Present value of an ordinary annuity:

= \$5,000 x [1 - (1 + 0.05)-20] / 0.05
= \$5,000 x [1 - (1.05)-20] /0.05
= \$5,000 x [1 - 0.37689] /0.05
= \$5,000 x [0.62311] / 0.05
= \$5,000 x 12.462, or \$62,311.05

Future value of an ordinary annuity:

= \$5,000 x [(1.05)20 - 1] / 0.05
= \$5,000 x [2.6533 - 1] / 0.05
= \$5,000 x [1.6533] / 0.05
= \$5,000 x 33.0660, or \$165,329.77