Read

Ordinary Annuity

This lesson explains what an ordinary annuity is, how equal payments made at the end of each period build up over time, and how to use the future value and present value formulas to solve real-world savings, loan, and investment problems with worked examples.

What Is an Ordinary Annuity?

An ordinary annuity is a series of equal payments or deposits made at regular time intervals, where each payment happens at the end of each period. Think of a monthly car loan payment, a regular contribution to a savings account, or the equal payments on a mortgage. As long as the amount is the same every time and it lands at the close of each period (end of the month, end of the year, and so on), it is an ordinary annuity.

This idea connects directly to the compound interest ideas you use when you look at a bank account that earns interest over time. Instead of one lump sum growing on its own, an ordinary annuity tracks many equal deposits or payments, each one growing (or being discounted) for a different length of time.

The Timeline: Payments at the End of Each Period

A timeline makes the "end of period" rule easy to see. Notice that no payment happens at time 0.

Future Value of an Ordinary Annuity

The future value tells you how much all the deposits are worth once the last payment is made, including all the interest earned along the way. The formula is:

$ FV = PMT \cdot \dfrac{(1+i)^n - 1}{i} $

Here, $ PMT $ is the amount of each payment, $ i $ is the interest rate per period (written as a decimal), and $ n $ is the total number of payments.

For example, suppose you deposit $100$ at the end of every year into an account earning $5\%$ per year. The graph below shows how the future value grows as the number of deposits increases.

Future value of a $100 ordinary annuity growing at 5 percent per period.

After $10$ deposits, the balance is noticeably larger than $10 \times 100 = 1000$ because earlier deposits have had more time to earn interest, exactly the kind of pattern you see when planning long-term income and savings goals.

Present Value of an Ordinary Annuity

Sometimes you need to know the value today of a stream of future payments, this is the present value. It answers questions like "how large a loan can these monthly payments afford?" The formula for the present value of an ordinary annuity is:

$ PV = PMT \cdot \dfrac{1 - (1+i)^{-n}}{i} $

This formula discounts every future payment back to today's dollars and adds them all together. It is the equation lenders use to figure out loan amounts from a fixed monthly payment, and it is the same equation behind the "formula for present value of annuity" that shows up in personal finance and business math.

Worked Example

Suppose you plan to deposit $200$ at the end of every year for $6$ years into an account paying $4\%$ annual interest. Find the future value.

Identify the values: $PMT = 200$, $i = 0.04$, $n = 6$.

Substitute into the formula: $ FV = 200 \cdot \dfrac{(1.04)^6 - 1}{0.04} $.

Calculate $(1.04)^6 \approx 1.2653$, so $ FV = 200 \cdot \dfrac{0.2653}{0.04} \approx 200 \cdot 6.633 \approx 1326.60 $.

After $6$ years of deposits and interest, the account holds about $1326.60$, even though you only deposited $6 \times 200 = 1200$ in total. The extra $126.60$ came from interest.

Ordinary Annuity vs Annuity Due

An ordinary annuity pays at the end of each period, while an annuity due pays at the beginning of each period. Because annuity due payments start earning interest one period earlier, their value is always a little higher than an ordinary annuity with the same payment, rate, and number of periods. Watch for wording like "at the start of the month" or "in advance" as a clue that a problem is describing an annuity due instead.

Where Ordinary Annuities Show Up in Real Life

Ordinary annuities appear whenever equal payments occur on a fixed schedule: car loans, mortgages, RESP contributions, and regular retirement account deposits. Understanding the formula helps you plan a realistic budget by seeing exactly how much a repeated payment or deposit is really worth over time.

Related lessons

On Grade 8